Cortes, or rather his men, apparently either scuttled them (that is, broke holes in their bottoms) and then ran them aground to be broken up by the sea, or possibly just ran them aground, it's not clear which. The idea that they were burned doesn't really appear in literature until the latter part of the 18th century, and starts popping up in sort of "heroic" literature (poems in particular) about Cortes that seem to have borrowed fairly heavily from ancient literature. We know of ancient heroes who (at least legendarily) burned their ships -- the one with the best parallel to the legend of Cortes is probably Agathocles of Syracuse, who burned his ships upon landing in North Africa to strengthen the resolve of his men.

This is a good accessible article on the subject, if you have JSTOR access: https://www.jstor.org/stable/335707