。
She walked up to me and asked my name.
Literal
She [topic-は] my side [up to-まで] came-over [and-て] my name [object-を] asked.
やってきて is the te-form of やってくる, a compound verb literally 'come [doing]' — built from やる + くる, but functioning as 'show up, come over, arrive.' 私の側まで ('all the way to my side') uses まで to mark the extent of the approach — emphasizing that she came right up to the speaker, not just nearby. 尋ねる ('to inquire, ask') is more formal than the everyday 聞く ('ask'); the choice between them shifts the register. The whole sentence has a measured, narrative quality.