Because he's worthy of it.

The fact that the Vision can lift the hammer, in contrast to all the other heroes (apart from Thor, of course) is an important clue to convey the Vision's actual spiritual purity and superiority. Remember that the Vision was created in a collaborative process by the Avengers, taking Tony's benevolent AI J.A.R.V.I.S, putting it into the pure mind gem (now cleaned of whatever evil spirit was in it that's now in Ultron), putting that into a perfect Vibranium-enhanced body and putting it to life by Thor's very powers.

To me the Vision seems to be the embodiment of the ancient ideal of a perfect mind in a perfect body, one that combines the advantages of a mechanical intelligence with the concerns and soul of an organic lifeform and brings compassion for his (undoubtedly inferior) human fellows, in contrast to his much less modest and unidentifiedly-spirited brother Ultron. He thus seems to be the ideal creation of a humanity that strives for posthuman evolution. One could say in his second try Tony Stark (with the help of an actual (demi)god and some magic alien artifacts, though) overcame his role as merely a "Modern Prometheus" and didn't just create sentient life, but a new god in itself.

And the fact that Vision is able to wield Mjölnir to me doesn't come from the fact that he is "just a machine", as Tony and Steve argue jokingly at the end, but from the fact that he is superior to us all not only in his body but even more so in his mind. And this I think is the attitude that Mjölnir is attracted to, as already shown in the first Thor movie and also in this one when Steve nearly achieves to lift it (which was not because he is the strongest, but because he is the noblest at heart). In fact, Tony's and Steve's theories about Vision being "only a machine" at the end even support the interpretation that this is not the case by the mere humorous way they are presented in. Of course their childish excuses are not the reason for why The Vision achieved what they couldn't. In fact his pickung up the hammer is a perfect and to the point illustration of Vision's superiority and immediately clarifies that to the viewers, a symbol that has been explicitly set up by the earlier scene (with the whole gang trying to pick up the hammer) for exactly that purpose. Or as the apparent expert on the intricacies of Mjölnir has to admit himself at the end:

Thor: He can wield the hammer, he can keep the Mind Stone. It's safe with the Vision and these days safe is in short supply.

So yes, Vision does get the power of Mjölnir and Thor with everything connected, including the abstract notion of "being worthy". But what makes Vision so extraordinary as part of his "worthiness" and what makes it seem as if he doesn't get that power (i.e. without the nice special effects described in Dr R Dizzle's answer) is that he doesn't want it anyway. He doesn't strive for this power and he doesn't care to rule over Asgard, not because he can't as "a machine" or because he doesn't have a soul, but because his mind and motivations are above such "worldly" considerations. His light-handed and downplayed way to hand over the hammer to Thor and his easy ability to carry it in battle convey both his worthiness and his not caring about this at all. He is a god without a god-complex.