These lines speak to accepting what happens, and being willing to rebuild when things are torn down, in order to be a man.
When Rudyard Kipling says, "If you can bear to hear the truth you've spoken twisted by knaves to make a trap for fools," he means that sometimes, even when you speak the truth, others will change your words to hurt others or convince others of untrue things. He is saying that you have to be able to live with this: it is inevitable. You have to be able to overcome the negative feelings that might arise from this and move on with your life.
Kipling goes on to say, "Or watch the things you gave your life to, broken, and stoop and build ’em up with worn-out tools," which means that when things fall apart, a man fixes them. He doesn't simply mourn the damage that's been done: he is willing to look at his life's work after it is damaged, accept it, and then move to fix it.
Each of the lines in the poem describes someone who is a man, in Kipling's eyes: someone who is responsible and mature. It's not just about being male; it's about having the persona that comes with being a grown and mature man. When someone can fulfill all the qualities that Kipling has laid out—including the two quoted above—then they are able to claim that they're a man.