Learn German: Er hängt das Bild ans Regal. - He hangs the picture on the shelf.

We use the accusative case here because the sentence describes an action moving toward a destination ("Where to?" or Wohin?). ans is short for an das, which is the two-way preposition an plus the neuter accusative article for Regal. If the picture were already hanging there without moving (a fixed location, "Where?" or Wo?), we would use the dative case: am Regal (an dem Regal).

Why use the preposition an? Doesn't auf also mean "on"?

Yes, but German uses different words for "on" depending on the surface! You use auf for horizontal surfaces (like resting something flat on top of a table or shelf). You use an for vertical surfaces or boundaries. Since he is hanging the picture, it is going on the vertical side or edge of the shelf, not resting flat on top.

Does the verb hängen always describe a movement or direction?

Not always! hängen is a special verb that can describe either an active movement (putting something somewhere) or a state (being located somewhere). Because he is actively moving the picture to a new spot, we treat it as a direction and use the accusative case. If we just wanted to describe the state of the picture hanging there, there is no movement, so we would use the dative case: Das Bild hängt am Regal.

Can I just write an das instead of ans?