基本はアスカがシンジをいじめる。で、たまにキレたシンジがアスカを襲う。その時のアスカはやけにしおらしい。

。、。。

Basically, Asuka bullies Shinji. Then, sometimes Shinji snaps and goes after Asuka. At those times, Asuka is strangely meek.

Literal

Basic [topic-は] Asuka [subject-が] Shinji [object-を] bullies. Then, sometimes snapped Shinji [subject-が] Asuka [object-を] attacks. That time [possessive-の] Asuka [topic-は] strangely meek.

A character-dynamics summary, clearly about Evangelion's Asuka and Shinji. キレる ('to snap, lose one's temper') is a slangy verb for sudden anger. しおらしい ('meek, demure, subdued') describes someone acting unusually docile — a word with a slightly old-fashioned or literary flavor. やけに ('strangely, excessively') signals the speaker finds the behavior out of character. The で linking sentences is colloquial narrative sequencing.