Why is MA not provable from ZFC?

There are so many reasons, and which one you want to use depends a lot on what you may consider as simpler and which consequences of Martin's Axiom you may be familiar with.

1. Adding a single Cohen real will add a Suslin tree. Add a Cohen real to any model of $sf ZFC+lnot CH$, and you will have a model in which there is a Suslin tree, and therefore Martin's Axiom fail.

2. Adding $aleph_3$ Cohen reals to a model of $sf ZFC+GCH$ will satisfy that $sf CH$ fails, but at the same time, every real is either in the ground model, or Cohen generic over the ground model (i.e. generic for a countable forcing). Taking Random, Hechler, or many other type of c.c.c. reals which are not added by Cohen forcing, we have that these forcings have $aleph_2$ dense sets in the ground model, but since $aleph_2<2^{aleph_0}$ in the generic extension, we should be able to find these type of reals.

   This can be done with Random reals, or other type of reals where we know of easy "c.c.c. reals which are not added".

3. As bof suggests, since Martin's Axiom imply that ${sflnot CH}to 2^{aleph_1}=2^{aleph_0}$, all we need is to negate that. Easton's theorem tells us that this is quite easy to do.

4. Cichon's diagram of cardinal characteristics becomes somewhat trivial under Martin's Axiom, all but $aleph_1$ will be equal to $2^{aleph_0}$. So any result separating the characteristics (e.g. small non-meager sets, or small non-measurable sets, or small unbounded families, or small dominating families, to name a few) will imply that Martin's Axiom fail. This area is studied extensively, and in fact many of the basic results have simple constructions: adding $aleph_1$ Cohen reals will add an unbounded family of size $aleph_1$ to any model of $sf ZFC$, in particular one where $sf CH$ was already failing.

And there are probably other approaches one can take. This is by no means an exhaustive list.