お預かりします is “get used to it.” よろしくお願いします is “get used to it.”

The essence of “get used to it” is finding those sentences that surround you in everyday life and learning how to use them/what they mean. It is beneficial to break the phrases down and see how they work, but try not to expend too much brain power or else you’ll be one of those freaks explaining the origin of saying “bless you” when someone sneezes. (Demons make you sneeze, of course, thus making you in need of a blessing. Duh.)

Saturday I was watching TV and saw one of these:

I’m sure you’ve all noticed it at some point. At different points in a show, generally just before a commercial break or the end of the show, the phrase 提供 (ていきょう) pops up on the screen with some company names underneath it. Then the voiceover announcer says 「ご覧のスポンサーの提供でお送りします。」

I believe ご覧 refers to the audience looking at the actual company names written on the screen, so literally “the sponsors you (honorably) see.” That combined with 提供 gives us “the contribution of the sponsors you (honorably) see.”

お送りします is the humble honorific of 送る (おくる), so that’s the television channel itself doing the sending, giving us, “We (humbly) send you (this show) via the contributions of the sponsors you (honorably) see.” Woo.

In English we’d probably say something like, “Brought to you by State Farm Insurance – Like a Good Neighbor, State Farm is there!” That or "Support for Show X is provided by Chevy Trucks – Like a Rock."