The phrase "to puff out" means to inflate or expand something, often by blowing air into it. It can also mean to become proud or self-important, as if one is blowing up their own importance.
Sentences with «to puff out»
What could be causing the star to puff out dust in fits and starts? (newscientist.com)
However I am so much more patient and willing to play mega blocks for an hour after sneaking a little magic dragon puff out the kitchen window. (crappypictures.com)
This is an example of Warren Buffett's concept of «cigar butt» investing: Someone may have tossed it on the ground, but you can still get a few good puffs out of it. (alephblog.com)