Very roughly:

One “transaction” on the ETH blockchain costs $X.

Therefore they are batching up 10 “transactions” at a time and executing them together as a single “transaction”. This way it only costs $(X*0.1) for each “transaction”.

"Centralized finance is bad, and blockchain solves that. Except blockchain doesn't scale to millions of clients, so we are going to introduce a centralized mechanism that solves that problem, while hiding the fact that we violated our founding principal of decentralization (by overwhelming you with crypto-jargon)."

Without knowing exact implementation details for Base, here is a summary:

Coinbase will run a network which developers can use for storing game state (think in-game items), and the transactions that this network runs are regularly sent to Ethereum a snapshot and anyone can provide cryptographic proof if they see that Coinbase submitted invalid data and force the network to not accept that bad data (though that process probably not implemented yet and its unclear how the economics would work if Coinbase can't be financially punished for fraud like typical layer 2 validators).

This provides good security at the cost of decentralization, but you aren't held hostage by the centralized actor (Coinbase) like you'd be in a traditional centralized system.

This might mean that these games will have the ability to use each other's state and have any level of interoperability that each developer chooses, and we don't have to fully trust that Coinbase isn't cheating the users or devs.

Because this is a network that's secured by Ethereum, assets can also be moved from this network to the main Ethereum network if users want.

A monad is a monoid in the category of endofunctors.

Optimism is an Ethereum optimistic rollup, which means it batches transactions off-chain and then publishes the batches on-chain periodically. Optimistic rollups are said to be 'optimistic' because the assumption is that all the transactions follow the rules and there's a 7-14 day challenge period where anyone can submit a fraud proof and challenge the history of the chain. If fraud is determined to have occurred, then the sequencer who batched the transactions is penalized and the fraud verifier is rewarded. The benefits are two-fold: 1) The off-chain transactions are cheaper because the gas fee is split among many participants instead of one 2) Users of the rollup can exit the rollup back to Ethereum at anytime ensuring full custody of their funds, even if the rollup was malicious or if the rollup halted.

Optimism has open-sourced their optimistic rollup stack which allows anyone to fork Optimism and create their own rollup. This is what Coinbase has done. So Coinbase will now have a rollup of their own where they will be responsible for the sequencer (batching transactions) and anyone can deploy EVM compatible applications, like all of the other Layer 2 rollups currently in production. So Uniswap for example may deploy to Base, which will allow Coinbase users to easily use cheap defi.

A rollup means instead of putting transactions directly on Ethereum they go to a Coinbase-managed chain that later commits those transactions and their results to the Ethereum chain in a cheaper way than executing them on-chain directly. "Optimistic" means there is a period where people can prove to the chain that Coinbase lied about the transaction results and after that period it's assumed they're correct.