When something goes "without a hitch" it means something happened without a problem. When horses get hitched, it means they get tied to a wagon. A carriage and a wagon are similar, they are both vehicles that carry something to move it from one place to another. A carriage will typically carry people, while a wagon carries people or things. The joke is playing on the word hitch and its different meanings. 
The first time he put the horses on the carriage it went without a hitch .