I read about zkSnarks and it says that it will increase the stability by 30x by conducting the verification off-chain. What does off-chain mean here? Nodes will still have to use their computation power to conduct the verification, right? Does it mean that it will save space by not considering each and every verification step as a transaction to save space? When learning about lightning network it was pretty clear that there is going to be a fixed size pool and users will be able to change their share and finally publish it to the chain when both parties agree. What happens in this case? am confused.
The term off-chain is very generic. See this What are offchain and onchain Transactions? for a definition.
Now zk-snarks is a tool to shows proofs without revealing data.
There were some attemps to use zk-snarks to scale transaction processing, for example On-chain scaling to potentially ~500 tx/sec through mass tx validation.
One variant of plasma is to store on-chain the root of the merkle tree of plasma transactions. And use zk-snarks to show a proof the tree root was updated correctly. You save blockchain space because the proof is of much smaller in size than all the transactions.
These plasma transactions are processed in a separate plasma chain, hence they are off-chain. In case of malicious operator with the off-chain data you should be able to prove the operator actions were incorrect.
In normal blockchain, each full node executes all the transactions to make sure they are valid.
With zkSnarks node may just verify a proof that somebody else has verified the transactions. Such proof may be much shorter than transactions themselves, and verification of such proof may be much easier than executing all the transactions.