Unusual features in animals (and the narwhal tusk certainly must qualify) can most often be explained by advantages provided for foraging, defense against predators, adaptation to extreme environments or in connection to reproduction. It is common for the three first explanations that one will expect to see the feature in both males and females, whereas the opposite is the case when it comes to reproductive specializations. I'm sure people can come up with exceptions, but as a general rule it is useful. Hence, the most likely explanation for the (male) narwhal's tusk is that it gives them advantage in reproduction, analogous to antlers in male deer, and also the tusks seen in males of most beaked whale species. Exactly what advantage a large tusk gives a male narwhal is difficult to find out, but if they engage in head-butting, as is not uncommon among male mammals when fighting over females or territories, it seems fair to assume that early proto-narhwal males, with one or more slightly protruding tooth had an advantage, as they could injure their competitors with this tooth. Just as seen in the paired scars many male beaked whales have, which closely match the distance between the tusks in their lower jaw. Once such males with a protruding tooth were around, it is easy to see why there would be a strong selection for having the longest tusk in a head-butting competition!
Alternatively (or in combination), tusks could be a form of 'honest signal', useful for females to judge the health status and strength of the male. It must be a very large handicap for a male narwhal to have a large tusk. Not only does it make swimming (and turning) very difficult (imagine swimming with a 2 m pole stuck to your forehead), it also warns potential prey well in advance about the arrival of the whale. If a male has a large tusk, it therefore must mean that it has been able to not only survive but also grow the long tusk and therefore would be expected to have good genes.
If one wants to explain the narwhal's tusk with one of the alternative proposals, a lot of additional explanation is required. Especially important is the question of why females do not have tusks, if they are so useful to males. And yes, occasionally female narwhals do have a tusk, but they are tiny and unlikely to be functional in the same way as the males'. Because females don't have a (large) tusk, it is difficult to believe that it has a function related to the Arctic environment, or that it has an important role as a sensory organ. The walrus is a very good example of the opposite. Here the tusk are clearly useful in foraging and to make it possible for the animals to get up on the ice, which is why also females have them.
Explaining why narwhals are the only whales with a tusk is close to impossible and also less informative. Evolution does not result in the most optimal solutions to all problems. Rather, it works by the principle of improving something that already works a little bit in the right direction. This often results in many different solutions to the same problem, for example the above-mentioned tusks in beaked whales.