。
She makes an effort to keep up with what's happening in the world.
Literal
She [topic-は] world [in-で] happening things [to-に] not-falling-behind keeping-up [so as to-ように] is-making-effort.
世の中で起こっていること means 'things happening in the world' — a relative clause nominalized with こと. 遅れずについていく is 'to keep up without falling behind' — 遅れず (without being late, classical negative) + ついていく (to follow along). ~ように努力している means 'is making an effort to' — ように marks the goal and 努力する means 'to make effort.'