I regret nothing is a line that is usually paired up with a GIF of an animal or character that is dancing, spinning, or doing something visually intense.
It is used to express one’s reluctance of regret or remorse after doing something that is immoral, stupid, or ethically questionable.
The phrase itself is conveying a historical background.
In 1960 SS Officer Adolf Eichmann is recorded to have said the sentence while waiting for his trial on court.
In 1992, an episode of The Simpsons also featured the quote, uttered by a character falling down a bridge.
Before the meme was born, the origin of the quote was disputed in a thread on Metafilter in 2005.
Later, in 2009, image macros with animals started to use I regret nothing as well.
The meme made the biggest impact when it was used in relation with the GIF Spinning Disco Chicken after 2011.
The combination of the two memes inspired a lot of people to put the quote on various other GIFs in order to utilize them as reaction memes.