For this explanation, lets say that everyone on the pool has miner type SuperMiner10 that the pool 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 miner is expected to find - on average - a 1 Difficulty share for every 2^{32} hashes it does. If the miner is mining at 10,000 Difficulty, then it is expected find - on average - one 10,000 Difficulty share for every 10,000 x 2^{32} hashes it does. After sending many 10,000 Difficulty shares to the pool, each 10,000 Difficulty share sent proves you have done the work equivalent of 10,000 x 2^{32} hashes. Also, 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 each.

If your SuperMiner10 is mining at at 5,000 Difficulty, it will find - on average - 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 each.

Of course if someone gets a 'new' 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.

Why Does the Pool Do This?

The pool doesn't know what hash rate your miner is actually doing, but shares do prove what equivalent 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 equivalent hash rate your miner is doing.

Now-a-days with Asic-Boost in most miners, even your miner doesn't know what equivalent hash rate it is doing, it must use the same process of estimating the hash rate based on the Difficulty of the shares it finds, otherwise it would report a much lower hash rate.

If you wish to read up more about that Proof of Work, google 'BitcoinProof of Work' or 'BitcoinPoW'.