People who lack fortitude whine, repine, and are anxious any time there is a threatening situation that presents itself. These people are the faint-hearted, the "cowards" to whom Caesar refers. Since they quake in the face of authority, of truth, of integrity, of danger, they, therefore, "die many times" as, in their pusillanimous reactions, they surrender their authenticity.
However, the individual of integrity does not surrender his values in situations that are threatening because he has fortitude; he is authentic. Therefore, he only dies when his body stops breathing whereas the coward has seen himself quake, or "die," over and over.
The Founding Fathers of this country lived this ideal as presented by Caesar [e.g. Thomas Paine: "Give me liberty or give me death!"; Thomas Jefferson's "It is the right and the duty of the people to revolt" when a government becomes too oppressive], but nowadays, our politicos live as the "cowards" of Caesar's speech.