。
She gradually began to understand.
Literal
She [topic-は] gradually understand-began.
次第に (しだいに, 'gradually, by degrees, little by little') is a slightly literary adverb — its everyday spoken counterpart is だんだん or 徐々に. The 次第 root carries a 'sequence, order' sense (cf. ~次第 'as soon as,' 手紙の次第 'the contents of the letter'); 次第に extends that to 'in order, step by step, gradually.' ~始める (here ~はじめる) attaches to a verb stem to mean 'begin to V' — a productive aspectual auxiliary alongside ~続ける (continue) and ~終わる (finish).