How did the Alien and Sedition Acts destroy the John Adams presidency?

The draconian Alien and Sedition Acts (1798) were fatal to the Adams presidency because they appeared to confirm what his Republican opponents had been saying about him for years. They'd always said that President Adams wanted to make himself into a kind of king, that he wanted to establish a tyrannical central government that would infringe Americans' inalienable rights and take away their liberty.
That's precisely what the Acts appeared to do. As well as making it easier to deport so-called aliens, more ominously they placed severe restrictions on speech critical of the government. Whatever the intentions behind these controversial pieces of legislation, there's little doubt that they were used primarily to attack Adams's Republican opponents.
Although Adams had reservations about the Acts, he signed them into law anyway. In doing so, he was effectively signing his own political death warrant, practically guaranteeing that he'd lose the next election, which he did.