I overslept and was late for school.

Literal

I [topic-は] overslept-[sequential-て] school [to-に] was-late.

寝坊する ('to oversleep' — literally 'sleep-laziness') chains with て-form to 遅刻する ('to be late'). The て-form here expresses sequential cause-and-effect: oversleeping caused the lateness.