I wrote aileyle and got this one wrong. It's hard to hear the difference for me when the audio is played normally, but if it were to be played slowly, I could tell that it said aile ile. They both mean the same thing, it's just that aile ile was said in this audio and not aileyle.
Because "ailesi ile" would mean "with his family", specifically.
"Aile ile" means "with the family" i.e. some family, not necessarily his own.
"Aileyi ile" is incorrect and doesn't mean anything. The word "aileyi" is in the accusative case so if you have the family as the object of a sentence, that's fine.
E.g. Köpek aileyi koruyor - The dog is protecting the family
Now some languages do have something called a "comitative" case, where a noun is treated as being "with" another, but English does not and neither does Turkish. Instead both nouns are expressed in the nominative case, as subjects, and joined with "with" - "ile".
Note that "aileyle" - "with the family" or "ailesiyle" are not a different verb case, they are simply less formal contractions of "aile ile" and "ailesiyle", because they mirror how the words sound when spoken (quickly).