。。
Please leave it to me. I'll take care of it.
Literal
I [to-に] entrust [please-ください]. I [subject-が] care [object-を] see.
Two short sentences. The first (僕に任せてください) is the standard polite request 'please leave it to me' — 任せる ('entrust') with the to-marker pinning the speaker as the recipient of trust. The second uses 面倒を見る, literally 'to see/look at the trouble,' the fixed idiom for 'to take care of, to look after.' Commonly applied to caring for children, pets, plants, or work tasks. The が in 僕が highlights the speaker as the one volunteering, distinguishing them from anyone else.