"Life only avails, not the having lived." The word "avail" means "to use to one's advantage." So this saying could be paraphrased as: Life itself is what's important, not the fact that you lived it. You could say that this is an argument against despair that leads to suicide.
"Power ceases in the instant of repose." The word "repose" means "to rest" or "to sleep." So a paraphrase might be: "No one is powerful when asleep."
"What I must do is all that concerns me, not what the people think." This one shouldn't need any paraphrasing for understanding, but you might restate it as: "It doesn't matter what people think, I'll do what I have to do."
"Travelling is a fool's paradise." Paraphrase: You have to be crazy to like to travel!