If a person gets an annulment so that he can get married again, how is this not going against Jesus’ words that the two become one that no man can put asunder?

An annulment does not “put asunder” (Matt. 19:6) a marriage. Rather, it recognizes and declares that, based on sufficient evidence, a true marriage never actually came into existence. What might have appeared to have been a marriage was not truly one. Thus, there is no marriage to put asunder.