There's a bit of speculation here, since there is no blatant in-universe explanation nor Word of God to fully demonstrate the reason for this.
You may want to take a look at the answer to why a soul bonded to a suit of armor can move, because the explanation here is pretty similar. Essentially, keeping in line with the series' representation of eyes being windows to the soul (a common trope), basically: Sight is not necessarily handled biologically.
Here's basically how this works: If an average person loses his sight (as does happen in the series), he cannot see, because there is no channel of vision connected to his soul (for either eye). However, Alphonse, Barry, and the other soul-bound armors do have this channel of vision, just as there is the ability for them to move their steel fingers (with, of course, the channel being in opposite direction).
The reason what he sees does not collide with what his body sees is because his body is not transmitting its sight to a soul. It has no conscious mind to connect to its sight. As for his sleep, because he has no biological mechanisms, they do not need to rest, but his conscious mind still remains active because it is affixed to his soul (imagine the way we dream).
You're probably thinking, "Oh, hurr hurr, mister clever, there's still no physical organ to transmit the vision data!" And you'd be absolutely right! There is nothing which suggests that there is a physical entity able to transmit visual data from the real world to the armor seal, whereas there is a physical entity able to move his limbs. It's not even known that only armor with helmets can see; in fact, when Barry is reduced to a single plate of armor, he appears to be able to see his body standing over him.
So to put it simply, your answer is: Alphonse and other soul-bound armors see because it's necessary for the story. We can loosely explain it, as I did above, but there still is no legitimate, solid reason why this should be possible.