Step-by-step explanation:
Heathcliff showed Nelly with his behavior how is he really feeling about everything and when she rejected to take a letter from him, he was so angry and devastated so she decided to do that.
He was using threats to persuade her and he was warning her that he will do anything, he will show up with pistols or he will harm someone. That was something that impressed Nelly and she decided to give that letter to Cathrine because she also wanted to save everyone from Heathcliff.