In English
Lets say everyone on the pool has miner type SuperMiner10 that the pools sets the difficulty to 10,000
When your miner is mining at 10,000 difficulty, it means it sends shares (worth 10,000) to the pool as proof of the amount of work it is doing over the time between submitting each share to the pool.
1 Difficulty is 2^32 hashes or a bit over 4 billion.
A 10,000 difficulty share
proves you have done work equivalent of 10,000 x 2^32 hashes
You can't fake it.
So e.g. if you have 2 miners sending 10,000 difficulty shares - then you would send, on average, twice as many shares as another person with only one miner.
Thus when rewards are calculated, you'd expect (on average) to get twice as much reward as the person with only 1 miner.
NOW, if you change the difficulty, it's expected to NOT effect your reward.
The relationship between the difficulty and the expected amount of shares is linear.
If your SuperMiner10 is mining at 20,000 difficulty, it will find (on average) half as many shares over a given time period vs mining at 10,000 difficulty.
So since each share is now worth
twice as much, the overall result will be the same - half as many shares worth twice as much.
If your SuperMiner10 is mining at 5,000 difficulty, it will find (on aveage) twice as many shares over a given time period vs mining at 10,000 difficulty.
Again since each share is now worth
half as much, the overall result will be the same - twice as many shares worth half as much.
Of course if someone is mining with a SuperMiner20 that hashes twice as fast as the SuperMiner10, then they also expect to get twice the reward.
The pool will set double the difficulty so they find roughly the same number of shares as the slower miner, but each share
proves you've done twice the work of the slower miner.
And lastly, before anyone gets some weird idea about trying to earn more by messing around with the difficulty ...
Nope you can't, coz people thought about that long ago and ensured you can't do that with proof of work.
So why do we do this?
Coz the pool has no idea what hash rate your miner is actually doing, but shares do
prove what hash rate your miner is doing.
Otherwise a person could mess with their miner and get it to say it is mining at 1000 times faster than it really is.
Again shares do
prove what hash rate your miner is doing.
Edit: if you wish to google about that proof - it's called "Proof of Work" or PoW