In English one would say "I like to learn their functionality" or "I like to learn how they function" , saying "I like to learn their functioning" is something you would never hear
"Functionality" has a slightly different meaning again, closer to "function" than "functioning" (the purpose rather than how it works internally).
"I like to learn their functioning" may be stilted, because there are more straightforward alternatives available, but it's a correct sentence that accurately translates the French. You might certainly hear someone say for example "I study the functioning of the 19th-century British legal system."
Personally I went with "I like to learn how they work." :)