The person who first made this claim is the person that the continents are named after: Amerigo Vespucci.

Vespucci was an Italian born and raised in Florence, who participated in two Portuguese expeditions to South America in 1499 and 1502. Vespucci created various accounts of the travels in which he staked the claim that the discovered landmasses were not the eastern fringes of India/Asia, but rather a "New World" - the name he gave the discoveries in a letter to the Medicis of Florence. The letter was published in Latin under the title "Mundus Novus" (= New World).

Vespuccis accounts and most importantly the "Mundus Novus" letter led Martin Waldseemüller, a German cartographer, to label the New World "America" after the Latin variant "Americus" of Vespuccis first name used in the letter.

For the "how did they know" part - I can't help you with that, but I am interested in it myself so I hope someone can chip in on that.