According to a professor in this article on phys.org it's because the population exploded when they wiped out all the forests. What caused the population explosion? Was it the development of a new technology that enabled them to consume at a faster rate? Why would deforestation have made them go extinct instead of wiping out only some of them, lowering the consumption rate enough to prevent their extinction?
They didn't. The (much reduced) population is still there. They went through a very bad time with wars and starvation, and records of the time were lost with the last people who could read the writing system. Wikipedia says:
Polynesian people most likely settled on Easter Island sometime between 700 and 1100 CE, and created a thriving and industrious culture as evidenced by the island's numerous enormous stone moai and other artifacts. However, human activity, the introduction of the Polynesian rat and overpopulation led to gradual deforestation and extinction of natural resources which severely weakened the Rapa Nui civilization. By the time of European arrival in 1722, the island's population had dropped to 2,000–3,000 from an estimated high of approximately 15,000 just a century earlier. European diseases and Peruvian slave raiding in the 1860s further reduced the Rapa Nui population, to a low of only 111 inhabitants in 1877.
When I was there in August last year I met a very old woman at one of the sites and when I signed the guest book she got very excited and spoke at length in a language that was neither English nor Spanish. I managed to catch "maururu Canadense" many times. Maururu is Thankyou in Tahiti and presumably also there (Polynesian words are similar throughout the Pacific) and Canadense is doubtless from the Spanish. My guide told me that in the late 50's or early 60's a group of Canadian doctors came and helped the natives who were very ill and being treated poorly by Chile, but more importantly took blood samples and demonstrated that the people who live there today are descended from the original settlers who built the Moai. This was a huge deal to her. She made a point of thanking and welcoming us. Our doctors gave them their heritage back.
Now, as to the collapse reasons, early visitors reported large forests, but those disappeared. To this day there are almost no trees outside of town there, just some eucalyptus that were planted during the 20th century sheep ranching times. There are almost no spots on the island where you can safely go into the water. There's a harbor in the town and a beach. There is no reef around the island. This means the people couldn't be supported by fishing, gathering seaweed, or anything other than the land unless they built a boat to go out into the water. Without trees they couldn't make boats. They had to change their houses from wood to stone - and very small stone houses they were indeed. Opinions vary on the reasons for the forest collapse - some say "the ignorant savages" cut down every tree without realizing they needed them to survive. I don't believe that myself. If you've stood in the quarry looking at the moai half-carved, you understand how much effort and organization was required to make them and move them, this was a culture that knew things and made things happen. Someone fed those artisans. Clearly people didn't make their own statues, they commissioned them. They worked in teams. Even a group madness for bigger and bigger statues would not have led to accidentally cutting down every tree on the whole island and then saying Ooops. Other theories say rats from visiting ships ate the nuts of the trees meaning new ones didn't grow. Pollen records show that the tree population crashed over a very short period of time. The loss of the trees led to the collapse of the human population, but the exact reason for the loss of the trees may never be known.