“Caerse” means to drop oneself, that is, to fall. “Se” (“itself”) refers to the cup. “Le” refers to the old man. Literally, this means the old man’s cup fell, but in English we’d say “the old man dropped the cup”. That would literally be “el viejo dejó caer la taza”, but we don’t usually say that because in Spanish, it would imply the man dropped the cup intentionally. “Se le cayó” means he dropped it by accident. The same with “se le rompió” — (literally “it broke itself to him”) means “he broke it (by accident)” in English. In Spanish, “lo rompió” would mean that he broke it on purpose, which is not the case here.