How does 'bloom' work?

As @Kevin stated, this can be easily confused with graphical effect Bloom without knowing the context in which the term is being used.

In the context of shooter games, the term is known as Reticle Bloom and is very similar to the term Weapon Spread. Reticle Bloom and Weapon Spread both relate to the loss of accuracy when firing, but the difference is where the inaccuracy is lost.

Weapon Spread, like this answer states, is where the bullet looses accuracy in relation to the distance of its target. A good example for this would be a Shotgun.

Reticle Bloom is where the loss of accuracy is centered on the gun and is in relation to how fast the gun is being fired, and the target Reticle becomes larger if not ceased.

Halopedia's definition for this is

reticle bloom, this is where sustained fire will decrease the accuracy of the weapon being fired and also increases the size of the reticle

Using a gif borrowed from giphy of the game Halo Reach, you can see the reticle become larger after continual, rapid shots making it more difficult to aim

Bloom is the spread of the bullets in the aimer, or the four white markers in the center. The Bloom of a gun is its random chance of hitting the target straight on. So, the more bloom a gun has in fortnite, the less chance the bullet will fly perfectly straight, no matter if you are crouching or not moving.