Here’s a nice video how the double stage rotary screw air-ends of Ingersoll Rand works.
In this compressor element, there are two sets of rotary screws: 2 big ones (low pressure stage) and two smaller ones (high pressure stage).
It is said that:
They provide increased reliability by reducing the compression ratio required for each stage resulting in lower bearing thrust loads and increased airend life.
Sounds logical, but they probably come with a big price tag, and they have more moving parts, which can be a disadvantage.