Please help me understand this

I'm reading Sapiens and the author is talking about Buddhism. I understand for the most part the basic philosophy that people suffer by trying to chase happiness and avoid pain. The author writes, "most people wrongly identify themselves with their feelings, thoughts, likes and dislikes. When they feel anger, they think 'I am angry. This is my anger.'... they never realise that they are not their feelings, and that the relentless pursuit of particular feelings just traps them in misery." The premise that chasing these feelings enslaves them makes perfect sense to me. However I can't seem to make sense of the claim that these feelings are not really theirs? Like how does ones feelings and thoughts not represent them?