Good question, and one that has undergone heated debate amongst scientists. They did go extict at a time of great climate change, when we were leaving the last ice age around 10,000 years ago, but they managed to survive previous rounds of climate change. The most probably reason they went extinct is because of humans hunting them to extinction thousands of years ago, although climate change and disease are likely contributing factors.
You’re right that mammoths had thick, wooly fur coats. Sadly this didn’t save them!
We don’t know for certain what caused the mammoths to become extinct. One theory is that the climate became warmer and this caused the mammoth’s surroundings to change from grasslands to woods and forests. This could have made it difficult for the mammoths to find the food they needed, causing them to starve and slowly die out.
Another theory suggests that us humans hunted the mammoths to death. It may seem strange to think that people could have such a huge effect on a whole species of animal but we know that humans hunted the dodo bird into extinction just over 200 years ago.
Seeing as mammoths had survived climate changes in the past, a combination of habitat loss and over-hunting seems the most likely answer. However, some people believe that the mammoths were wiped out by an infectious disease.