Some lines stay inside the show that produced them and some end up in general use by people who never watched it. The difference lies in the structure of the line rather than in the popularity of the programme.
The line has to work without the scene
A joke that depends on knowing a character's history cannot travel, because carrying it requires carrying the history too.
Lines that escape tend to describe a recognisable situation rather than comment on a fictional one, so they can be applied by someone with no knowledge of the source.
This is why the phrases that spread are often not the best jokes in a script. They are the most portable ones.
Length is the second filter
A phrase has to be repeatable from memory after a single hearing, which puts a practical limit of a few words on it.
Rhythm matters as much as brevity. Lines with a strong stress pattern survive repetition, and flat ones do not.
Delivery contributes, but only up to a point, because a phrase that requires a specific performance to work will fail in the mouth of an ordinary speaker.
Repetition inside the show does the initial work
A catchphrase is usually established deliberately, appearing weekly until the audience anticipates it.
Anticipation is the mechanism. Once viewers can predict the line, saying it themselves becomes a small pleasure rather than a quotation.
Sitcoms built around recurring characters exploit this directly, which is why the format produces more escaped phrases than serialised drama does.
The meaning drifts once it leaves
A phrase in general use quickly loses its original sense, because most people encounter it from other speakers rather than from the source.
Each retelling strips a little context, and after enough hops the line means whatever the situations it is used in have made it mean.
Original fans often find this irritating, since the version in circulation is a simplification of something they experienced with its full setting.
Why the source stops mattering
Once a phrase is common enough, attribution disappears. People use it having never identified where it came from.
At that point it functions as ordinary vocabulary, and the show has lost any control over it, including any commercial benefit.
Attempts to reclaim it, through trademark or through a revival built around the line, generally read as heavy-handed, because the audience now regards the phrase as theirs.