To "bring out" a quality means to make that quality stronger or easier to see, taste, etc. In the example above, the speaker says that toasting the nuts "brings out their flavor". That means that it makes the nuts taste better by increasing their natural flavor.
Other examples of "bringing out" something include:
Difficult situations like this really bring out the best in people.
This is a chance for you to go wild and bring out your inner child for a day or two.
Money brings out the worst in us.
"Bring out" is also used quite often to talk about flavors in food, like in the example at top.