。
She walked fast to catch up with everyone.
Literal
[topic-は] everyone [to-に] catch-up [trying-to-ようと] fast walked.
追いつく ('catch up to, overtake') takes に for the person/thing being caught up to. The volitional 追いつこう plus と forms ~ようと ('intending to, trying to') — a more abstract motivational frame than the explicit ~ために. 速く is the adverbial form of 速い ('fast'), modifying 歩いた ('walked'). Note the difference between 速い (fast in motion) and 早い (early in time): the kanji distinction matters and learners frequently mix them up; both read はやい. The whole sentence sketches the small drama of falling behind a group and hurrying to rejoin.