The term "Latin" America grew out of the fact that the languages spoken in these countries grew out of Latin. Which languages were used?

Mostly Portuguese and Spanish, but also a little French.

Explanation:

The term Latin America is used to refer to both Central and South Americas. Spanish is most largely spoken, unless we take only South America into consideration, in which case Portuguese wins by a small margin. The French Guiana has French as its official language. There are, however, non-Romance languages spoken in Latin America, such as English and Dutch, but by a minority.

Portuguese, Spanish and French belong to a group called Romance languages along with Italian and Romanian. They are languages that derived from Vulgar Latin and still share basic vocabulary - though with variations in pronunciation and spelling - and grammar structures.