Absence means not being present .
grow fonder refers to caring/ loving /liking more.
The above phrase thus means that One starts feeling more affection towards someone/something in the absence of them.
History :It was first used in the Thomas Haynes Bayly's (1797-1839) song Isle of Beauty as :
"Absence makes the heart grow fonder, Isle of Beauty, Fare thee well!"
The phrase was been commonly used henceforth.
1) Their relationship got better while working at different places. They proved that absence makes the heart grow fonder.
2) If one believes that absence makes the heart go fonder, then ask them why long distance relationships never work.