To "eat the dirt" is one of the colorful euphemisms riders sometimes use for falling off. Around a barn you might well hear stories where somebody says "I ate the dirt." or "I really thought I was going to end up eating the dirt."
Other colorful euphemisms for falling off include:
Going "out the door" - sometimes specified as the "front," "side" or "rear" door.
Doing or making an "involuntary dismount."
"Chewing gravel" is sometimes used in the west, with similar implications to "eating dirt."
Jumpers may mention "jumping the fence without the horse" - which is self-explanatory.
And if it's the horse's fault?
You got "thrown," "tossed," "dumped," "bronced," "dusted," "grassed," or "busted off."
A "wreck" is sometimes used by western (and some American English riders) to refer to a situation where both horse and rider fell, or where injuries resulted from a bad situation such as a horse getting stuck or taking an uneven or unstable trail. (I've also heard that some cowboys use this to refer to a fall caused by the horse being crazy, but not the ones I know).