It basically comes down to a question of the unit of selection.
From the common viewpoint, in which natural selection is seen as acting on individual organisms, it's almost a tautology that the organisms favored by selection are those that maximize their own reproductive fitness. Thus, the possibility that some organisms might engage in acts that help another organism at their own expense may seem like a paradox, or at least a puzzle in need of explanation.
One solution to this puzzle is offered by the gene-centered view of evolution, where selection is viewed as acting on genes, with organisms being merely convenient (and usually, but not always, cooperative) collections of genes that (usually) reproduce together. From this viewpoint, it is not at all surprising that evolution might favor genes that cause an organism to help other organisms, provided that there's a statistical tendency for those other organisms to also be carrying the same gene.
Other mechanisms for the evolution of cooperation do also exist. For example, organisms with sufficiently advanced cognitive capabilities may indeed engage in reciprocal altruism, where they help others only if those others have shown themselves willing to help them in exchange. Such exchanges, being mutually beneficial, do indeed help both participating organisms, and are thus selected for even at the organism level. However, to persist (in the absence of gene or group level selection effects), they generally need some form of enforcement and/or learning mechanism — if cheaters can keep receiving help without ever helping anyone else, they'll do better than the cooperators and eventually displace them.
Also, in some situations, an organism acting simply to help itself (e.g. by modifying its environment to be more suitable for itself) may also end up helping other organisms that just happen to be nearby. In such a situation, selection indeed favors cooperative behavior in isolated organisms by default, with the evolution and persistence of "freeloaders" (who spend less effort on improving their surrounding, relying instead on others to do the work) being only possible as long as there are sufficiently many cooperators around.
Of course, none of these mechanisms are mutually exclusive. Indeed, the presence of some gene-level selection effects tends to be unavoidable in any realistic situation involving cooperative interactions; it's hard to come up with a setting in which no two interacting organisms ever share genes, and obviously the fact that two organisms share a common genetic predisposition to cooperate makes them more likely to do so when they meet.
(However, one not-so-realistic theoretical limit case where that does happen is that of an infinitesimally rare cooperative mutant invading an infinitely large and well mixed population. The popular use of this simplifying limit assumption in classical models of evolution may be one reasons why cooperative behavior is so often seen as something remarkable and hard to explain by evolutionary theorists.)