What does the yellow "Joke's on You Batman" graffiti mean that we see on the suit in Batman's cave in Batman V Superman: Dawn of Justice?
It's a reference to the comic book arc Batman: A Death in the Family
"A Death in the Family" is a Batman comic book story arc first published in the late 1980s which gave fans the ability to influence the story through voting with a 900 number. "A Death in the Family" ran in Batman #426-429, published in 1988-1989. The story was credited to Jim Starlin (script), Jim Aparo (pencil), Mike DeCarlo (ink), Adrienne Roy (color), and John Costanza (lettering). Covers were illustrated by Mike Mignola.
The conclusion of the story was
the death of Jason Todd (second Robin) by the hand of Joker.
And that costume belongs to Robin (As you can clearly see the R in the upper right chest) and referred to that mentioned incident in above spoiler block.
It was debated a lot which Robin that costume belongs too. Refer http://comicbook.com for the whole debate. But Dick Grayson rumor has been debunked. And it has been confirmed to be Jason Todd in deleted video from WB but Zack Snyder Says Dick Grayson's Robin Was Killed.
Even Suicide Squad confirms that theory:
According to ScreenCrush, in the quick flashes of profile text that pop up during the introductions to the members of Suicide Squad, one detail in Harley Quinn’s list of attributes and accomplishments is rather interesting. A line of text says that she is an accomplice to the death of Robin.
tl;dr: It's a deceased Robin's suit; the assumption is that Joker killed Robin sometime in the past and it has haunted Bruce ever since.
The movie does not explain the grafitti. It's meant to be part of a yet-untold backstory of Batman. It's possible that part of the story will be told in future movies, possibly in a Batman solo movie.
However, we can make pretty good guesses at what's going on based on what we do see in that movie.
For starters, we know that before Dawn of Justice, Batman was a vigilante in Gotham for some time, long enough to be come a "legend". We also know that he gave up that job, for unstated reasons, and become bitter and depressed. It wasn't until Superman arrives that Batman decides to go back to crime fighting.
If we look at the suit:
you can see a stylized "R". This was a Robin suit, but we never see Robin in the movie. The strong implication here is that Joker murdered Robin, then vandalized the suit and left it for Batman to find. It was the death of Robin that turned Batman into the dark, violent, cynical version of Batman that we see most of the movie.
Bruce makes a number of comments during the movie that support this idea, though the name "Robin" is never actually used, nor are any of the names of Robin characters, such as "Jason Todd" (the most likely Robin for this to be), "Dick Greyson" or "Tim Drake".
Later, in Suicide Squad, some of this is confirmed: the introductory graphic for Harley Quinn claims she was arrested and charged with being an "accomplice to the murder of Robin". Since we know that HQ was attached closely to Joker until her arrest, this again implies that Joker was the one ultimately responsible.