>I I take M = {p_0, ..., p_m} to be the maximal affine independent subset of A, then M should span A, but what can I do with this information?

Show that it does in fact span M. It may be easier to argue by contradiction: if it does *not* span M, then there is some point x in M not in the span, so show that {p_0,...,p_m,x} is also independent, contradicting maximality.

>Also why can I "choose" a _finite_ maximal subset?

You're in R^(n), a space of finite dimension, so there is an upper bound on the size of an independent subset. Do you know what this upper bound is?