This is a misunderstanding. The goal of cryptocurrencies is not to process transactions. The goal is to order transactions. The order is the hard part. (Notice that if I have one ETH, and I submit a transaction to send alice an ETH and to send Bob an ETH, alice and bob are both very interested in the order - only one transaction can go through, and which one is determined by which one comes first in the chain).
Fundamentally the order doesn’t really matter, as long as it doesn’t change retroactively. Once you have agreement on the order, anyone can process the transactions and know they’ll get the same final result as anyone else.
I’m not sure about this specific one, but most “rollup”s use the main chain for agreement on the order, and then outsource the actual processing to someone outside the chain. Some use cryptography from verifiable computing, so the off-chain person can also submit a proof that their processing is right and there’s no possibility of fraud. (the proof can then be verified extremely quickly.)
Fundamentally the order doesn’t really matter, as long as it doesn’t change retroactively. Once you have agreement on the order, anyone can process the transactions and know they’ll get the same final result as anyone else.
I’m not sure about this specific one, but most “rollup”s use the main chain for agreement on the order, and then outsource the actual processing to someone outside the chain. Some use cryptography from verifiable computing, so the off-chain person can also submit a proof that their processing is right and there’s no possibility of fraud. (the proof can then be verified extremely quickly.)