Remember how 取る means "to take," and 出す means "to take out"? Well, when you combine the two, the meaning is still to take out.

Compared with 出す, which has a wide range of meanings, 取り出す is limited to the action of taking something out from somewhere. It feels a bit more descriptive than 出す — like you're reaching into a bag or a drawer, taking something in your hand, then pulling it out.

The readings come from the individual words 取る and 出す. Put them together and you should be able to read this word too.