They didn’t. Latin dances originated in a lot of different places in South America and are heavily influenced by Afro-Caribbean rhythms from the booming slave trade and trans-Atlantic travel of the 1500’s-1800s.
Samba originated in Brazil in the very early 1900s: Samba - What Is It? - Orlando Ballroom Dance Party Portal
Salsa doesn’t have a single point of origin but Cuba likes to take the credit for it: Salsa - What Is It? - Orlando Ballroom Dance Party Portal Salsa includes influences from Puerto Rico, Haiti, Africa, and even a little bit of European country dance styles. Mambo is also Cuban, but today’s Mambo is basically the Salsa on a different beat.
Tango comes from Argentina: Tango - What Is It? - Orlando Ballroom Dance Party Portal
Merengue hails from the Dominican Republic but Haiti likes to claim credit for it: Merengue - What Is It? - Orlando Ballroom Dance Party Portal
Cha Cha is genuinely a Cuban dance, having been created by a Cuban composer who invented the music that people eventually developed a dance for: Cha Cha - What Is it? - Orlando Ballroom Dance Party Portal
Bachata comes from the Dominican Republic: Bachata (dance) - Wikipedia
Rumba is a Cuban dance, but it also has some differences with today’s rumba/rhumba in the US. The *music* came from Cuba, and a dance was made up to go with the music, but the 2 versions danced today are American Standard (which was invented in the US) and International Standard (which was invented by a French instructor in London). Rhumba - Wikipedia
Bolero is a dance that has two separate styles and two completely separate and independent origins - Cuba, and Spain, with the Cuban version being heavily influenced by other countries like Puerto Rico and Mexico: Bolero - Wikipedia
Paso Doble is usually categorized as a “Latin dance” when you watch the TV competitions, but, ironically, the partner dance is French (based on Spanish military marches & bullfights), and then adopted by Spain and Portugal: Pasodoble - Wikipedia
And then there’s Jive, which is classified as a “Latin dance” under International dancesport categories, but Jive originated as Lindy Hop in New York at the Savoy Theater by a primarily black community and was later codified by Arthur Murray and other ballroom studios to make it easier to teach, and also to compete in. This led to the development of several different sub-categories of Lindy, and the competition version which is classified as a “Latin dance” is called Jive: Swing Dance - What Is It? - Orlando Ballroom Dance Party Portal