Does he say “you could keep it in the quote of the phrase” at the end? Thanks.

"I think I couldn't keep it in, [to coin a phrase](https://dictionary.cambridge.org/dictionary/english/to-coin-a-phrase)."

I think he says "I couldn't keep it in — to coin a phrase" Of course, to "coin" a phrase means to invent a new one, but it's often used ironically after using a very well known saying as well.

The other comments here are correct, he is saying "I think I couldn't keep it in, to coin a phrase." Basically he's saying that he couldn't resist from boasting/describing his romantic experiences in his biography. He begins to say after, "I had to let it get [out]" David Niven is incredible. Loved him in the original Pink Panther, Casino Royale (playing a parody of James Bond) and in Murder By Death (here parodying detective Nick Charles from the wonderful Thin Man series)