These are 2 separate phrases, Yunah. 'Fall out' is a phrasal verb which means to argue and stop speaking to someone 'On my account' is a phrase which in this context means 'because of me'. So: Don't argue and stop speaking because of me This phrase "on (someone's') account" can also mean to benefit the 'someone' Example: 'don't bother on my account' means don't bother to benefit just me