It means you don't want to do something or you don't agree with something. It can have a negative feeling or a disapproving feeling.
I want to go walk in the part tonight.
I'm not down with that. (I don't want to do that.)
You can say the opposite, "I'm down with that", if you want to do it.
Let's go to the movies.
I'm down with that. (Yeah, I want to do that.)