"Entrarle" is a colloquial term (usually used in Mexico, I do not know if other countries also use this term with this meaning) referring to do something eagerly and without fear, determined to achieve what you want. It can apply to a job, games, tests, women. For example:
Si consigo le reunión con él, le entro con todo = If I get a meeting with him, I will tell him everything I want to say (without fearing the consequences).
Voy a entrarle a las pruebas finales = I will do my best to pass the final tests.
So, your sentence "Cuando me gusta una chica, la entro con confianza, sin nervios..." could be translated as "when I like a girl, I approach her with self-confidence and relaxed" (literally "not being nervous").
I hope this helps.