The little amount of body hair humans have don't seem to be of much use for keeping warm. Our Simian cousins on the other hand sport thick furs.
At which point during the species evolution and why did humans lose their fur?
This is an interesting question, and there are a number hypotheses available to explain this phenomenon. The short answer (as far as I can say after my literature search) is that we don't know the answer for sure. The long version follows below.
The main problem with all these hypotheses is that though they may have a valid point, a definitive hypothesis hasn't been demonstrated yet. All these hypotheses are listed in the first reference, with a ton of additional references in the paper. I will summarize the different hypotheses only briefly here; for details please read the paper. The second reference is also interesting.
The point when this process started was after the last common ancestor of chimpanzee and humans when our two lines diverged. Chimpanzees are our closest relatives and still have hair coats, so this process must have started afterwards. Analyses of MC1R mutations (which affect pigmentation) suggest that human have been hairless for at least 1.2 million years. See reference 3 for details on this.
Aquatic ape hypothesis (also known as aquatic ape theory): This hypothesis states that modern humans spent some time semi-aquatic. In this lifestyle fur does not insulate very well and this led to an evolutionary disadvantage.
Cooling hypothesis: This hypothesis states that the modern humans lost their hair when they left the forests and started to live on the savannah. There they got too warm with their fur and finally lost it. This hypothesis sounds logical at first thought, but has a few problems. Naked humans have two problems: during the day they collect quite some heat via their skin, which requires them to have a cooling system (we sweat to cool us), and during cold nights we need a lot of internal energy to keep our temperature stable. Insulation however works both ways (not only keeps the warmth in, but also can keep heat out), and indeed some savannah monkeys have denser fur than their relatives in the forest. The cooling system will also lead to water loss and potential dehydration.
Hunting hypothesis: This hypothesis states that humans started becoming carnivores and therefore had the need to run for extended periods of time to hunt for meat. This leads to thermal overload; without a fur they can release this heat much faster to avoid overheating.
Bipedality hypothesis: This hypothesis assumes that modern humans, walking on two feet, have a lower direct influx of solar radiation. This lowers the chance for overheating.
Allometry hypothesis: Allometry states that species which get bigger during evolution, do not have all their organs getting bigger at the same rate. The hypothesis states that humans got bigger, the number of hairs stayed the same, and finally got lost.
Clothing hypothesis: One of the more illogical hypotheses, this assumes that hair loss wasn't critical for humans as they managed to make themselves clothes.
Vestiary hypothesis: A mix of the cooling and the clothing hypothesis: Here it is assumed that hair loss is beneficial due to more cooling and that the negative effects of too much loss of heat were counteracted by the invention of artificial insulation.
Neoteny hypothesis: Neoteny denotes the retention of of juvenile physical characteristics in adults. Humans are characterized as retarded in development (compared to the apes) and mature more slowly. Additionally, it is assumed that some characteristics (here loss of hair) of juvenile or fetal apes are maintained.
Carrion-eating hypothesis: Another strange hypothesis, this assumes that man as a messy eater (like vultures and condors, which have naked necks) have an advantage of being hairless.
Sex-related hypothesis: This hypothesis assumes that hairless skin is more sensitive to stimuli like touch, temperature and pain. This would allow more sensitive social contacts, especially between man and woman, but also between woman and child. Pubic hair is not explained by this hypothesis.
Adaptation-against-ectoparasites hypothesis: Here it is assumed that naked humans are less prone to harbour ticks, fleas and other parasites which hide and live within fur. Apes have great social rituals of removing these within their group.
The short answer is that no-one knows. That no-one knows, along with a lot of popular interest, has led to wide ranging speculation. Some of that is well grounded, but much of it is not. Some has been published within relevant journals subject to peer review, whilst others have been published independently for the lay public. Despite the extent of popular interest it is not a subject of study that attracts large numbers of academics or a lot of research funding, so there may not be enough experts to keep the more egregious speculators in check.
I think "Why?" is a question that brings to light more of what we don't know that what we do. We don't know if this peculiar evolutionary development was a consequence of specific mutations that resulted abruptly in occurrences of a furless variant amongst precursor hominids or if it was an accumulation of incremental changes via sexual and/or natural selection. I'd like to think studying DNA will answer that.
Furlessness probably can't be understood without reference to other changes, including to skin pigmentation, thickness, sweat and oil glands, follicular nerve supply and sensory sensitivity. Bipedalism too. And I don't think we can leave out the abilities of smart, tool using problem solvers to adapt behaviourally to overcome physical limitations, or that those kinds of solutions can go beyond mere compensation for disadvantages to become extraordinary advantages in their own right.
The most widely accepted explanation - that it is an adaptation for improved hot weather endurance - is predicated on an improved ability to perspire. I don't think it is known if being more sweaty preceded, followed or occurred together with truncated body hair growth, or if it one element of more wide ranging changes, such as with hypotheses invoking neotony/pedomorphism. When such an important advantage as hot weather endurance is dependent upon changes other than to hairiness, it's necessary to consider how and why they came about as well.
The little amount of body hair humans have don't seem to be of much use for keeping warm.
This raises a fundamental question of what the functions of body hair are. Whilst keeping warm is the function of fur in mammals that most immediately comes to mind to most people it is not the only function. I don't think the "why furless?" question can be answered through consideration only of the loss or diminishing of any specific functions; the myth of body hair being effectively functionless is what first prompted my interest in questions like this.
A widely overlooked observation is that modern humans gained improved sensory sensitivity through relative increase in follicular nerve supply, to the extent that W.Montagna ("Evolution of Human Skin", 1985) likens all human hairs, regardless of size or location, to the vibrissae - dedicated feeler hairs - of related apes.
Can we properly assess the relative advantages and disadvantages that arose from loss of fur if we do not properly consider the full range of functions of the hairs involved? Curiously, the hypothesis by D. Morris ("The Naked Ape", 1967) that losing fur improved our skin's sensory abilities, with consequences for social and sexual bonding, appears based on absence of hairs - presuming hairs to be inhibitors of touch sensitivity. To be fair, Montagna's observations that the increased follicular nerve supply has made them "the principle anatomical unit of skin sensibility" came after Morris. To be less generous the role of hairs as sensory receptors is self evident - unless his didn't work - and had made it's way into journals as far back as Darwin's day. (Can't find that reference to "every hair, with it's envelope, is a sensory organ" - a scan of a page rather than something searchable. Lost the link with a change of computer.)
I don't know that I've answered the question - more like pointed out that there is no definitive answer to this question; I would like to further discuss this, especially through engagement with people better informed than I am. Would this be a suitable subject for a Chat room? (Not sure I have the privileges here to start one).
Surely the simplest explanation is that hair is a "necessary evil"; a not terribly good compromise, but the best that evolution could come up with. Hair is very expensive biologically; it requires large amounts of energy both to produce and to maintain, and it harbors parasites. True, it keeps the animal warm when the environment is cold, but it also severely limits strenuous activity when it's hot. Lions and most similar predators can only manage short bursts of high speed running, but fur-less humans can run all day, chasing down large herbivores until they collapse from heat exhaustion.
However we're only able to do without body hair because, uniquely among animals, our high intelligence allows us to develop workable substitutes: fire, clothes, houses and so on. In short, the answer to the question: "Why, out of all the primates did humans lose their hair?" might simply be: "Because they could..." The "Aquatic Ape" hypothesis might better be re-worked as a consequence of developing hairlessness, not a cause. Being able to dry off rapidly after going into water (and if necessary put on dry clothes afterwards) made bodies of water much more attractive as a potential food source.
Essentially, losing our body hair arguably opened up new food sources to humans, ones that were particularly abundant in the fatty acids required for proper development of the infant brain. Better nutrition would have fostered natural selection for even greater brain development, eventually leading to modern humans.
Since there is only theories I will give my opinion. I think Fire played a huge part in losing our body hair 1.2 million years ago. Since the first fire was created 3.4 million years ago evolution had time to play it's part. Simply put, fire near a fury animal is way more dangers than a fire near hairless animal. And because fire gave it user many benefits to survive they couldn't let go of it just because it's was dangers and they adapted. Also, since fire gives heat it allowed us to stay worm without fur in the cold nights.