Not a historian, but work with a lot of antique firearms

Firearms have gone through several hurdles when it comes to being instrumental battlefield tools

The first major hurdle in the revolutionary/civil wars was accuracy. A smoothbore rifle slinging a smooth ball could veer wildly from POA to POI (point of aim to point of impact). To counter this, first they lengthened barrels, and then they began to add rifling to those barrels. Rifling is a basically to be thought of as train tracks for ammunition. There are grooves in the barrel that cause the round to spin at a desired rate to increase accuracy.

Fast forward to WW1, and you see the emergence of the first automatic weapons like the Madsen, Lewis gun, etc., you’re now in the age where accuracy is manageable thanks to shaped rounds, cartridges that control powder much more accurately than hand loaded muskets, and rifling on the barrels that shaped rounds are giving you accuracy much further than any point before. At this point, it was accuracy that needed improvement, it was fire rate. You see a lot of the worlds first machine guns debuting right before/during/after WW1, as well as the emergence of what would later become sub machine guns, such as the MP18 (video of one on my profile, as well as a Lewis gun and another weapon or two)

By WW2, the had the accuracy for single fire down, and the suppressive power of automatics pretty well dialed in. The problem was putting the two of them together. When shooting a fully automatic weapon, it’s very hard to put multiple rounds on target, accurately. Typically you’ll hit the first 1-3 rounds before muzzle rise and recoil have messed up your point of aim, and then you’re just trying to keep the rounds in the general direction

The thinking was by creating a burst fire weapon, it still allows for a very rapid rate of fire, but the bursts allow for much more accurately suppressing fire, as well as allowing for “more” rounds on target, as you’re fighting much less recoil out of a 3 round burst than by holding the bang switch

TL:DR, burst fire is the combination of accurate single/semi rates of fire and auto, to give as much accuracy and rates of fire as possible

Edit: grammar, sorry.