Cutting In Butter Or Shortening

What does it mean when a recipe says to cut butter into a flour mixture?

It means breaking up the butter or shortening into the flour.

For pastry, you break up the butter/shortening into smaller pieces so that the flour gets coated with some fat and the flour won't absorb too much liquid.
For a struesel topping, you cut in butter to get a crunchy crumbly topping.

There are pastry cutters you can buy to cut in the butter/shortening.
You can also use a fork to just break it up.

What I prefer to do is to work the butter into the flour or sugar/flour mix with my hands.
To do this, just stick your hands into the bowl with the butter and just rub it into the flour.
Make sure you end up with coarse crumbs.

For pastry, leave some of the pieces of butter larger and some smaller.

For a struesel topping, all the butter should be uniform so all the crumbs are about the same size.