The verb “become” can have a couple of different meanings but here is means to grow into or to change in someway.
If you become somebody else, it means you start to act like that person. You might start to behave similarly to this person or dress the same way as them or even start enjoying the same hobbies as them.
This often happens between friends or couples who live together or spend a lot of time together. One person might pick up certain habits of the other person. However, in extreme cases, one person might copy everything the other person like the way he or she speaks, acts, spends their spare time, etc. and become the other person.
So if somebody says “you become me” they mean that you are similar to him/her. That you have picked up some of his/her personality traits. Before spending all this time with this person, you were different but now you might behave like him or her, dress similar to him or her and have similar interests.
