Why do some people stand in the water to fish and not just stand on the bank of the river? They get wet this way.
Fishing in a river while wearing waders permits me to fish down stream and positioning my lures in the spot that is only possible while being in the water. This also helps me to avoid the many snags such as branches and tree trunks that are commonly found on the banks of many water ways. I can not tell you how many lures I have lost while fishing off the banks of rivers.
If you fish from the bank of a river you must cast your line upstream. eventually your line will float downstream and make its way towards the edge of water. Hopefully in a spot one can retrieve your equipment for another cast.
Fly fishing can only be done while in the water due to fact that one is moving the fly in in two directions, both in a forward and backward positions. It is the only way to avoid snags.
Since most people do not wear waders while fishing or venture into the water I have many more options as to where I want to fish. There is nothing more challenging than fishing on a bank with too many anglers in areas that are renown to be good salmon spots. With waders on I am able to go upstream 100 meters or so from the group and get the big ones all by myself. The spot I prefer to fish in has trees and shrubs on both banks and making it impracticable to access my fishing hole from the banks.
As a side note: Waders keep the leaches off!
They are not getting wet, they are getting farther from shore, and closer to the fish.
They are not standing on the shore, for all the same reasons a person in a boat is not standing on shore. Getting closer to the deeper water where the bigger fish are.
While you can't get as far in waders as you can in a boat, waders cost much less then a boat, and are easier to transport from home to the water then a boat. Waders will also allow you move around in water that it is to shallow or rocky for a boat.
In the image below the man is wearing chest waders. The stream is to small to navigate by boat. It would be impossible to fish from shore on the far side. A bit farther down stream it would be impossible to fish from the near side shore. With the waders on he can travel up and down stream, get to where the fish are without getting cold and wet.
River banks have trees making it hard to cast.
You don't have the angle to cast up (or down). Typically cast up and let the bait take a natural path.
For fly fishing it is the typically the only option.
In many places, the law requires it. A common access rule in many places is that public access is granted to flowing water, but only to the normal waterline. Access to the land above that water line is trespassing, requiring the fisherman to be in the water to access it. This is not always true, for instance some places prohibit being in the water to fish and some jurisdictions allow for also owning the river bottom which makes wading also trespassing without approved access.
The primary reason though is as stated, for correct presentation to the fish holding water. Wading should be done however only when actually needed to correctly access the holding water as excessive traffic does do damage to the stream and habitat.