Basically it enables people to write contracts with virtually no size limit in a modular and gas-efficient way.
Diamonds can be upgraded on the fly without having to redeploy existing functionality.
Now in a real scenario a diamond has many faces known as facets so just like that facets here are contracts that you interact with, upgrade etc through the diamond contract.
Diamond Cut is a style guide used to shape diamonds just like that to add, remove, modify any facet. We have a Diamond Cut Facet used to do the same so you can upgrade facets without any hassle.
Before diving into the UI and contracts, I will recommend to go through the EIP to have a complete understanding of the same.
For demonstration purposes, the current Facet linked with the UI is a DeFi which uses Aave Uniswap Market to leverage 2x by just depositing ETH, and it can be upgraded by redeploying it again(go through deploy.js in detail) after changes and choosing the right upgrade action in UI.
The upgrades are done currently by directly calling the Diamond Cut Facet.