。
She ran really fast, trying to catch up with the other members.
Literal
She [topic-は] other [genitive-の] members [to-に] catch-up [intent-と] very fast ran.
追いつこう is the volitional of 追いつく ('catch up'); paired with と, it gives the intentional reading — 'with the intent to catch up / trying to catch up.' This volitional + と frame captures the in-the-moment intention behind an action more vividly than the alternative ~ために ('in order to'), which leans toward deliberative purpose. メンバー is a katakana loanword from English 'member.'