Why do viruses typically affect only one or a few species?

Viruses are subject to the same evolutionary principles as their hosts. All genes are selected for based on their fitness- that is their ability to replicate in a given environment. The environment of a virus is the host organism itself. So, a virus that can effeciently and effectively target its host is likely to be selected for and reemerge in greater numbers in the next generation. The drawback, from the point of view of the virus, is that in order to be an effective virus, a virus must be target the very particular weaknesses of its host. Viral genes that target a specific host are likely to be selected for. Viral genes that may be more able to effect a greater variety of hosts but may not effect the actual host as well are unlikely to be selected for. So, viruses often become host specific, meaning that they can effect only a number of species or perhaps just one species. As they evolve to target one organism very well, they sacrifice the ability to target other organisms. Think of it as the difference between an AK47 and a sniper rifle. The sniper rifle "gives up" the ability to spray wildly, and maybe hit a greater number of targets, for the ability to precisely and effectively hit one single target.