The assumption is that there is not much room in the lane. When you pull in forward, especially with a longer vehicle, you need more room to swing out and get the front end aligned with the spot before you enter the space since the rear just follows the front turning wheels. When backing, you just have to get one of the rear wheels into position and then the front end swings out while pulling into the spot rather than before pulling in. It's way easier to pull out of the spot when you do this, too, because you can turn the wheel immediately, whereas when you're front in, you have to back almost all the way out before you can start cutting the wheel. Of course it also depends on how far past the rear wheels the vehicle extends as to how much it will swing out.

In tight fits you are much more able to park straight than if you parked nose in.more space for the front end to angle and get the back end where needs to be as well as space to move the front end back and forth to straighten out.

Nose forward you are pretty limited in sideways movement and need to do like 18 tiny 3-point turns to try and get it lined up good.

More maneuverability, being able to pivot around the wheels that are first to enter the parking space, kinda like the difference in results when going nose first into a parallel parking spot vs backing in.

The same reason parallel parking is done backing in.

So I've driven fire trucks and similar sized vehicles. If I'm trying to get the truck in a driveway and have two lanes to work with I can go nose first. I go into the lane opposite of the target driveway to swing the front end into the driveway. It definitely takes both lanes if you don't want to make a 100 point turn, Austin Powers style.

If it's a tiny road or only one lane then I have to back in. I approach by getting as close to the target drive as possible and then swing the nose away from that side of the road, lining up at a better angle when I start backing. This pic shows it well but you don't need nearly as much space irl. Your just go slower and cut the wheel harder. The back tire could be just a bit above and to the left the #3.

My point is you can get into a lot tighter spaces backing in. There's a reason why forklifts steer from the back. The truck in the pic should have backed in or started over.

If you back up with your front wheels turned all the way to the side, the back corner of your car barely moves. Mostly of the movement is your front end swinging to the side.

This can be useful when you need to make a sharp turn. It allows the back corner of your vehicle to make a very tight turn around the opening of the parking space.

Basically going forward, to turn the vehicle 90 degrees might take say 30 feet of forward motion. Going backward, it might only take 3 feet of “forward” motion to turn the car 90 degrees.

Much tighter turning radius for the end of the car opposite the turning wheels.

This is why a forklift’s steering control works by turning the back wheels not the front wheels. Allows that forklift to rotate around the front, without the front moving at all.

Forklifts have a more extreme version of this design since you can turn those wheels full sideways (and even a little backwards if you want), but the same principle operates in any vehicle with one set of turning wheels.