。
My hair has gotten too long.
Literal
I [of-の] hair [topic-は] long became-too-much.
長くなり is the conjunctive form of 長くなる ('to become long'), where 長く is the く-form of 長い. 過ぎる ('to exceed') attaches to verb stems to mean 'too much' — なり+過ぎる = 'became too much.' A compound verb pattern that attaches to any verb stem: 食べすぎる ('overeat'), 言いすぎる ('overstate').