The space around me is brightened thanks to the lightning.
The lightning lights up the space around me.
This area is lightened up by the lightning.
照明=lightning 、「この照明によって〜」であれば、"~~~thanks to this lightning" や”This lightning lights up the space around me."となります。「私の周り」は"the space around me"、また、指を指すなりして「この辺り」という意味で、"this area", "around here"となります。「明るくなる」="brightened", "lightened up", "illuminated"など。「照明によって私の周りはよく見える」であれば、"I see well around this area thanks to the lightning." "Because we have this lightning, I can see well around here."などです。