Quote Originally Posted by Lylandra
what does "Предоставь" mean litterally ?
Literally it means "give", "grant". Usually it is a literary word, used in certain contexts.

Your example is a set phrase, kind of. It is a shortened version of
"Предоставь/дай мне право/возможность заняться этим делом" ("give/grant me a right to deal with it")
Usually, nobody says that. People just say
"Предоставь это мне" ("leave it to me").