"He falls off his horse."
Translation:Il tombe de son cheval.
"De" can generally be translated as "from" or "of (as in, part of)", whereas "à" has meanings of "in", "to", "until", "belonging to"... For me, the easiest way was to think of "de" as having meanings associated with moving away from something, and "à" having meanings associated with being close to or moving toward something.
Take this case here: "He falls off his horse". I think of it as him literally moving away from his horse. Therefore, use "de."
Obviously there are exceptions and deviations (the first that comes to mind being "près de", meaning "near") that may render a different meaning, but I think taking this approach is a good place to start. Everything else comes with practice:)