The classics are Arendt Heytings "Intuitionism: And introduction", Stephen Kleene's "Introduction to Intuitionistic Mathematics" and Errett Bishop's "Schizophrenia in Modern Mathematics".
If you are a decent mathematician, or better yet, a computer scientist, Bishop's "Constructive Analysis" is really nice.
Each later author makes the case in wholly different way than Brouwer. Brouwer himself was actually kind of a fuzzy-yet-overly-picky thinker when it came to philosophical aspects, and he can be hard to take seriously. He also follows his own intuitions into mathematical endeavors that are completely barren, like the alternative models of continuity he proposed. So I prefer Kleene as the exemplar of a good, clean, productive Intuitionist. You can't claim recursive function theory did not go anywhere...
The argument that sold me, which I have never seen anywhere in print, is this: Mathematics is the oldest branch of psychology. What we study in mathematics is not some other world. It is the process of abstraction, as it has evolved in human beings. Which sets of rules are minimal, and which minimal rules fit together in constructive ways are psychological facts, not natural ones. Rules don't exist outside of minds.
Starting from that naturalized approach to mathematical truth, it seems obvious that we can't expect to avoid contradiction so completely that we can give everything a truth value. So does the idea that the only comprehensible infinities represent processes rather than entities. And the resulting modesty in other assumptions follow pretty directly from those two starting points.