Strictly speaking, lawnmowers are loud because they work by creating a vacuum, and vacuums make a lot of noise.
Looked at more broadly, however, it could be said that lawnmowers are loud because the purpose of mowing a lawn is to make as much noise as possible, so everyone will know that someone is mowing their lawn.
It’s possible to have a lawn without making a lot of noise. A homeowner could plant low-growing or slow-growing grasses, which stay at a lawn-like height while being mowed rarely to never. Homeowners could landscape with moss gardens, which are visually similar to lawns but don’t require any loud maintenance. Homeowners could install astroturf. They could use a virtually-silent reel mower. Or they could employ an automatic lawnmower, a small and quiet device which has been commercially available since the 1950s but which has never achieved widespread popularity.
Why have these quieter, simpler alternatives never caught on? In short, it is because the historical purpose of lawns is to be maintenance-intensive, in order to show off that the property owner can afford to spend their money and leisure time preventing grass from growing. This message is conveyed when neighbors see naturally-tall grasses staying perpetually short – but it is conveyed more effectively when everyone can see and hear the property owner actively maintaining their short grass.
Oddly enough, when some homeowners tried out automatic lawnmowers, their neighbors did not say, “Look at that smart guy enjoying a nice day while a robot does his yard work.” Instead, they said, “Look at that lazy guy lying in the hammock while a robot does his yard work.” For many people, what matters is not that the yard work is done; it’s that the homeowner does the yard work personally (or hires other humans to do it at obvious expense), and is observed to be doing it.
Noise harms neighbors. Lawns do not benefit neighbors. Those who care about their communities are increasingly embracing low-maintenance natural yards, or at least switching to quieter, less damaging ways of managing their turf.