What is the purpose of E-transition?

The purpose of ∈ transition is that, it allows a transition on ∈, the empty string. In effect an NFA is allowed to make a transition spontaneously without receiving an input symbol. The new capability doesn’t end / expand the clan of languages that can be accepted by finite automata, but it does give as some added programming convenience. Again NFA’s with ∈ transition which we call ∈-NFA are closely related to RE and useful in proving the equivalence between the classes of language accepted by FA and by RE.