As well as meaning "create a picture with a pen or pencil", the verb "draw" is also used to mean "pull".

According to Longman Dictionary, 'draw somebody to something' has the definition of 'to attract someone or make them want to do something' Example: Beth felt strangely drawn to his gentle stranger./// The festival is likely to draw huge crowds. Therefore, 'drawn to me' is something like 'attracted towards me'.