Rav Avigdor Miller (Tape: Happy Is He Who Ignores) explains that the idea behind this is that one should not pay attention to such thoughts of what other think of you. Why not? Because that will "eat you up".
So, “How happy is the man who hears and ignores.” Just ignore it. Because when a person doesn’t ignore it, it eats him up. And he gets emotional problems. So therefore a person has to think sensibly – think about Hakodosh Boruch Hu, how many good things He does for you in this world. Think about how lucky you are to have two good eyes – most of you have two good eyes. You have something to eat, you have a bed to sleep in at night. You have so many good things in this life, so try to enjoy them. ראה בטוב השם – “Look at the goodness of Hashem,” and then your character will often straighten out and most of your emotional problems will become nothing.
Rav Miller quotes the Gemara in Sanhedrin 7a, which writes:
It is good for a person who hears statements said against him and yet remains silent (טוביה דשמע ואדיש)
Rabbi Yisroel Greenwald in the article called "Tapping the power of Simcha" uses this Gemara and brings a story from a student of Novardok:
In Novardok, yeshiva students participated in deliberately humiliating behaviour, such as going to the bakery and asking for a box of nails, or wearing a tie made out of hay. A student of Novardok related to the author that the purpose of these exercises were not to "put yourself down," as is commonly thought by outsiders. The training, in fact, promoted the opposite; it gave the students the emotional freedom from the bogeyman of public approval. They discovered that the fear of embarrassment was actually much greater than the reality. This strengthened their confidence to do the right thing, oblivious to what others might think. The bravery and courage of the Novardok students was evident by their legendary ability to start yeshivas despite the risk of imprisonment by the Russian authorities.
A more comtemporary approach to this, can be found in the footnotes (footnote 47):
Rabbi Yehudah Mandel went to France to visit the Novardik rosh yeshiva, R. Gershon Liebman. Sitting next to the rosh yeshiva, he saw him take a handful of cornflakes and put it in his cup of coffee. When Reb Gershon noticed Rabbi Mandel's startled look, he scooped another handful of cornflakes, and, moving towards his guest's cup of coffee, asked, "Would you like some too?"
So yes, I suppose Yes Studios, which produced the series, made it their own and adapted that saying into their own way.