This old saying means that the lack of something increases the desire for it.
The Roman poet Sextus Propertius gave us the earliest form of this saying in Elegies:
"Always toward absent lovers love's tide stronger flows."
The contemporary version appears first as the title of an anonymous English poem in 1602. It wasn't until the 19th century that the phrase began to be used more widely, with Thomas Haynes Bayly's (1797-1839) song, 'Isle of Beauty', published after his death in 1850:
"Absence makes the heart grow fonder, Isle of Beauty, Fare thee well!"
The meaning of this song referred to England as he found that he missed his country when he was away from it.