Caer and its reflexive form, caerse, mean much the same thing and can even be translated the same way, typically as "to fall." But the reflexive form of some verbs, including caer, can be (but isn't always) used to suggest that action was unexpected or accidental rather than deliberate.
- La piedra cayó desde una altura de 800 metros. (The rock fell from a height of 800 meters.) Here the emphasis is on the nature of the rock falling, in particular the original height.
- Un sonámbulo se cayó desde el tercer piso. (A sleepwalker fell from the third floor.) The use of the reflexive form here emphasizes that the fall was accidental and/or unexpected.
The difference between these two sentences is primarily one of emphasis. Grammatically, you could have just as well used se cayó in the first sentence and cayó in the second. But the emphasis would have changed slightly.
- Caí a la piscina y el vestido se arruinó. I fell in the pool and my clothes were ruined.
- El otro día me caí a la piscina y mi celular conmigo. The other day I fell in the pool and my cell phone with me.
Is there any real difference in meaning between caí in the first sentence and me caí in the second? Not really. The difference again is one of emphasis. In the first one, the speaker is more matter of fact about his or her fall. In the second, the unintentional nature of the fall is stressed. Sometimes, the difference between caer and caerse is explained as the difference between "to fall" and "to fall down" or "to fall over."
A few other verbs can have a similar distinction between reflexive and non-reflexive forms. For example, "Salimos del juego" probably means simply "We left the game," while "Nos salimos del juego" suggests that the departure was in some way surprising, sudden or unexpected. Perhaps somebody got sick and couldn't stay.
Similarly, while "El autobús ha parado" may mean that the bus stopped at a regular stop, "El autobús se ha parado" might mean that the bus stopped unexpectedly, perhaps because of mechanical problems or an accident ahead.