What is this sleeve in this toilet tank?
All the answers thus far are correct, in that it's an (early) attempt to save water on flushing.
The important part I don't see in any other answer is that it's an early attempt, and they did not work very well. It's like watersaving toilet version 1.0 or 1.1, and watersaving toilet version 12.3 is on the market now.
If you'd like to save water while actually flushing waste effectively, upgrade the toilet to a current version. They work very well. That's a slapdash retrofit to a toilet that's designed to use a lot of water to flush well. The current ones are designed from the ground up to flush on less than half the water that probably uses, which is probably half what it uses without the plastic tube.