。
My sister insisted on coming with me.
Literal
Younger-sister [topic-は] me [with-と] together go [quotative-と] insisted.
言い張る ('to insist, to persist in saying') is a compound verb — 言う ('say') + 張る ('stretch/persist'). The と before 言い張る is quotative, marking what she insisted on. いっしょに ('together') is the standard adverb for doing things jointly.