Actually, it was more complicated than that. People cut down the trees, since they were used to carve out boats. They were the largest palms ever known, even larger than the Chilean Wine Palm - the largest non-extinct palm tree. Supposedly, the trunk of this tree could exceed 7 feet in diameter. The tree was not only the source of wood for their outriggers, but also a source of food and fruit.
New ones didn't grow because the rats gnawed and ate the seeds of the palm. Rats didn't cut down the trees, people did. You had to wonder what went through the guy's head who cut down the last tree...
Anyway, rats were introduced to the island by the people - probaby accidentally as is always the case.
Once the palms died out - along with all the other flora and fauna - the locals had no mobility. There was no more source of wood to build outriggers, so they could no longer trade with other islands, nor could they go out to sea. They were basically 'shipwrecked'.
Easter Island's End by Jared DiamondEnjoy. Happy New Year!