Don't quote me because I just learned this stuff like last week and don't fully understand it yet... but here's my best guess.

I think those vectors would span M then, given it is a 3x3 matrix. Meaning a particular combination of those vectors can produce any 3x3 matrix M.

To be independent, none of the vectors in a subset can be linear combinations of each other. If one of them is a multiple of another or linear combination of other ones, it is redundant and thus provides no new information - so it's dependant. In order to span the 3x3 matrix, the 3 vectors must be independent. But if it were a 2x2 matrix, I believe it can still span while being linearly dependent, provided only one of the vectors is redundant.