1-4 players (The 4 player cabinet requires the player to put the coin in the slot for the character they wish to play as. This isn't visible with emulation, so if you're going to play a 1 player game, and want to choose who you're going to play as, you'd be better off with a 2 player clone rom)
Button 1: Attack/Pick up weapon
Button 2: Jump
2, 1 to jump attack. There's a certain point early-ish in the jump height, where you do a different jump attack, but it's the same as pressing 2, Left/Right & 1, just not as controlled. 2 & 1 simultaneously is another jumping attack, and easily the most powerful move in a single players arsenal. If 2 players stand their characters close together, but don't press any buttons for a few seconds, a double team attack will start up, which either player should be able to control.
Food items, which restore health, need to be walked into to be consumed. Sometimes they, or weapons, are carried by other Simpsons characters you can just walk into to get. Trees will often drop an apple or an orange, if you attack them enough. Some "weapons" that can be thrown are living animals, like Snowball II. Sometimes the players character messes up during their attack combo, and does something to themselves. I don't know what triggers that, just an occasional bit of humor, but it can leave you open to attack. The interesting one is Marge's, which goes into an early production idea for the show that never came about, in which she shocks herself, and you see rabbit ears, which was intended to be the original explanation for why her hair is like that. On 2 occasions you get a button mashing bonus stage.
1991 is pretty early in Simpsons lore, so a lot of what a later viewer of the show, who didn't watch from the beginning, might expect, won't be there. As one example of what I mean, Sideshow Bob hadn't tried to frame Krusty for crime, or murder Bart yet, so he actually helps out at one point.
At the same time, this is easily the strangest choice for a license to try to adapt into a beat 'em up. Other licensed beat 'em ups already came from comics, action movies, or action TV shows, and actually make sense for the adaptation. This is the most out of left field idea, but it goes to show, a creative enough developer will make anything work.