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 posthumously in 1850:
"Absence makes the heart grow fonder, Isle of Beauty, Fare thee well!".