It's all to do with ullage in the fuel tanks. Newton's laws of motion mean that when a rocket is no longer firing and no force is being applied, the rocket receives no acceleration. It continues at its same velocity (if we assume a perfect model). The fuel in the tanks goes into free-fall, just as the astronauts do when they reach orbit. You've seen those soft toys going floating around the cabin when the Soyuz engines cut off.
The fuel also acts a bit like a passenger in a car when you brake hard. Instead of being pushed back in their seats when accelerating, they fly the other way, towards the windscreen. That's why we use seatbelts. The same for the fuel in those huge tanks. It goes whoosh; right to the other end of the tank, the farthest distance from the rocket motor. The problem is, you want it at the rocket motor end. If the fuel is not at the "bottom" of the tank the pumps spin on empty, and can burn out with nothing to suck on.
There are two solutions to this problem. Some craft have ullage motors which are little engines, sometimes just thruster jets or a release of compressed gas. You can see them in many videos of other craft that do not have this mesh. (Not to be confused with the pump exhaust nozzle, but that is another matter entirely!) It often looks like a small exhaust pipe near the engine. They can sometimes just be a small solid fuel engine, much like a firecracker. Their only function is to move (slosh) the fuel down to the pumps. The pumps start, and then the engine fires. Whoosh; rocket starts. Now all this fuel sloshing about can cause stability problems for the craft, so they design the tank with special anti-slosh baffles.
Here we can see those various features in the Saturn V second stage:
If you want to avoid the extra complexity, controls, fuel, mechanisms and just plain mass and money in your rocket, you can do it the Russian way. Just turn on the next stage while the previous one is still applying force. No need for ullage equipment. However, you have to make somewhere for all that hot exhaust gas (and its force) to go. The easiest way is to make a slight gap between the stages so you then get less of a baba-boom. It also has the nice advantage of pushing the completed stage away from the craft and back towards the planet, which is usually what you want. It might make the rocket taller, but it keeps the mass and complexity down, which is usually what you want.