We probably don't necessarily.
http://spanish.about.com/od/verbs/a/caer-vs-caerse.htm mentions: In many sentences they [caer and caerse] mean much the same thing and can even be translated the same way. But the reflexive form of some verbs, including caer, can be (but isn't always) used to suggest that an action was unexpected or accidental rather than deliberate.
I LOVE your answer!
We learners must remember that languages do not have one to one correlations. In theory, the way to express an idea in one language may take one word and the same idea in another might take ten.
I saw the example of Dejar caer meaning to fumble. That is to leave to drop... To let drop Which is very descriptive of fumbling really LOL!!!!
I still don't understand why 'ella se cayó' can't mean 'she fell over'? Is it that she has to fall down some incline (eg stairs) rather than tripping on a branch and falling over? I can imagine there might be a difference so i don't want to flag it yet, but i for my part would very very rarely say 'she fell down' in English unless it was followed by e.g. the stairs, the escalator, Mt Everest etc... it has also left me confused as to whether i could use this to say e.g. 'she fell off/from the roof' - would that be 'ella se cayó el techo'?
ella se cayó means the same as ella cayó, only the first one is more common. In english you can say "she fell" the same way as "she fell down", can't you? But which one do you use the most? the particle 'se' makes it more clear that she is the one who fell. If you want to express that she fell from somewhere, e.g. Mt. Everest (which is rather uncommon), you would say "Ella se cayó de el/del Monte Everest". No one say "ella cayó de (some place)"
To many Americans "she fell down" doesn't mean that she fell down off something, like the stairs. It's just that she fell to the ground (or floor or bed or whatever). The floor, etc., is lower, so she fell "down". So this must be American oriented translation. I don't know whether Spanish has another term for fell over. Perhaps this could be translated simply "she fell." I don't know.
She fell is accepted. Down is optional. In this case, the "se" is referring to herself. It's reflexive.
Hang isn't really the right word for caerse. http://www.wordreference.com/es/translation.asp?tranword=hang