“Hopefully that'll keep them occupied for a little while.”

You're riding back home from a vacation with your family. You have young children, and they're loud. You give them an iPod with a movie on it to watch so that they'll be quieter. You say to your husband:

Hopefully that'll keep them occupied for a little while.

hopefully

Start a sentence with "hopefully" when you're saying something that you hope will be true:

Hopefully we can finish up here by six or seven and go out for a beer afterwards.

that'll

"That'll" is a contraction of "that" and "will". People use it in casual spoken English. It's not very common at all in writing, though.

You should also learn the contraction "this'll":

This'll only take a minute.

keep (someone) occupied

To "keep someone occupied" means to give them something to do for a period of time. You keep people occupied so that they won't: 

• get bored
• cause trouble
• notice something bad that you're doing

For example, in movies thieves will do something to keep a guard occupied while one of them breaks into a place and steals something.

Parents often need to give their kids something to keep the children occupied while the parents are cooking or doing work at home.

You can also "keep yourself occupied" if you have a long period of time with nothing to do, like when you're retired:

He's been keeping himself occupied with some housework and a little reading.

for a little while

The phrase "for a little while" means "for a short time". "For a little while" is more common and natural than "for a short time" for a conversation.