“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.

There's lots of good answers, but I thought you might find this short article helps: https://www.lawlessspanish.com/grammar/verbs/accidental-reflexive/

Se= la taza Le = him/her cayó = verb(caer) meaning: It fell out of his/her hand or He/She dropped it. Se = the object Le = the person rompió.(romper) meaning: He/She broke it.

Le is "it", the cup. Se (itself) is joined with 'caer' (to fall) as 'caerse' meaning to drop, to let/allow to fall. One day the old man let go of the cup/ dropped the cup/ .. and it broke.