I never realized how difficult it was to explain slang until now!!
"Are you still down for tomorrow?"
That would basically be asking you if you were still willing to hang out tomorrow. (And "hang out" with you is slang for spend time with you.)
"I'm still down for tomorrow" means that I'm still available to hang out tomorrow. You can think of it as "available" or "wanting to" to spend time with the person you are talking to, I suppose.
I have no idea why it is called "down" but it's been widely used for the past few years.
Sorry if my answer didn't help. ><