What was Big Willy Shakes going for?
Armado is actually shirtless here. Scandalous, right?
His coin of the phrase "naked truth" is drawing on the idea that he is, well, naked. In fact, he's wearing wool right next to the skin (woolward) without any shirt or anything underneath. So maybe not so scandalous.
To put it another way, he's exposing the truth (and his chest along with it). He's giving us the truth, the whole truth, and nothing but the truth. And that's all this phrase really means. It tells us that there are no ifs, ands, or buts (or shirts, evidently) about it. That's the whole story. We've left nothing out. For Armado, it's clear to see that he's not lying because he doesn't have a shirt on.
John Lyly, a playwright doing his thing only slightly before Shakespeare, used this phrase in his play Euphues in 1578, but Shakespeare made it popular by giving it to us here.