Dear 100 Hour Board,
Why do Mormons always use the phrase "turning the time over"?
Probably for the same reason we say, "Nourish and strengthen our bodies," right before digging into doughnuts.
When you turn something over, you surrender control of it (see definition 4 here). When you turn the time over to someone, you put them in charge of the remainder of the time of the meeting. The example in definition 3 of this entry shows that this usage normal in non-Mormon settings. It's still kinda cliche, but it's not wrong.