(A) を (B) に任（まか）せる is a structure used to highlight that something is being ‘entrusted’ or ‘left up to (B). This can be thought of as a standard function of the る-Verb 任（まか）せる ‘to entrust’.
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(A) を (B) に任（まか）せる sentences, (A) を will identify the task, while (B) に will identify the person or thing that (A) is being left up to.
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私（わたし）は忙（いそが）しいから、あなたにペットの世話（せわ）を任（まか）せてもいい？
Can I trust you to take care of my pet since I'm busy?
先月（せんげつ）入社（にゅうしゃ）して来（き）たばかりの新人（しんじん）くんにあの重要（じゅうよう）なプレゼンを任（まか）せたの？
You entrusted that important presentation to a newcomer who just joined the company last month?
Fun-fact - Although this expression often highlights people that some task is being left up to, it is also frequently used when some eventuation or result is being left up to chance or fate.
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彼女（かのじょ）と付（つ）き合（あ）えるかどうかを神様（かみさま）に任（まか）せる。
I'm going to leave it to God to decide if I can go out with her or not.
あの会社（かいしゃ）に入（はい）れるかどうかは運（うん）に任（まか）せる。
I'll leave it to luck if I can join that company or not.