This indicates that the person described is unaware of his or her good fortune or is unaware of how difficult day to day life was before he/she was born. Typical usage: 'Kids today don't know they are born'.
To end on a muted note - most likely in a situation where one would have expected a more spectacular finish. This expression was coined by T.S. Elliot in his 1925 poem, The Hollow Men, which ends: This is the way the world ends This is the way the world ends This is the way the world ends Not with a bang but a whimper.