Are there ways to describe the Martin Axiom intuitively? I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking about this axiom?
Thanks in advance.

Since you mentioned in your comment that you're not familiar with iterated forcing, I'll try to explain a different aspect of $MA$, which I still hope can shed some light on this subject.
One way of understanding $MA_{\aleph_1}$ is in the general context of forcing axioms. If $\mathbb P$ is a forcing notion such that each $p\in \mathbb P$ has two incompatible members above it, then for every generic set $G\subseteq \mathbb P$, $G\notin V$. However, we would still like to find "quite generic" sets in the universe. More formally, let $\Gamma$ be a class of forcing notions. We say that $MA(\Gamma)$ holds if for every $\mathbb{P}\in \Gamma$ and a collection $\{I_{\alpha} : \alpha<\omega_1\}$ of dense sets, there is a filter $G$ on $\mathbb{P}$ that intersects all the $I_{\alpha}$. 
It turns out that we can prove the consistency of $MA(\Gamma)$ for non pathological classses of forcing notions. In this case, $MA_{\aleph_1}$ is simply $MA(\Gamma)$ when $\Gamma$ is the collection of c.c.c. forcing notions. By enlarging $\Gamma$ we obtain stronger forcing axioms, such as $PFA$ for the class of proper forcing notions and $MM$ for the class of stationary set preserving forcing notions. In this view, $MA_{\aleph_1}$ is just one level in a hierarchy of axioms that imply the existence of sufficiently generic sets.
