When my dishwasher drains, the water runs out the vent on my kitchen sink. Why?

It is partially clogged with food particles. The cap should unscrew. Take a look in there, it probably has some "nasties" in it. A baby bottle brush or something similar can be used to clean it out. Pour some HOT water in behind the scrubbing to help loosen that junk up.

Good Luck,
Stew

Also check under your sink where the dishwasher pipework connects. Make sure the pipe is straight up.

Good luck
LIn