L2 is, in my view, some of the most interesting research happening in computer science right now. The article above is not a great explanation--in particular, L2s are not off-chain as the article presents. The point of L2 is that it on-chain, inheriting the security and censorship resistance guarantees of L1.
To simplify: L2 is about creating a fast, high throughput state machine whose state transitions are verifiable on a blockchain. Blockchains, in turn, are about creating a uncensorable state machine that reaches global consensus.
So L1 achieves security, and L2 adds speed.
So why not just make L1 fast to begin with?
The strong guarantees of L1 rely on a lot of validators (on the order of ~10k+, worldwide, often on home internet connections) verifying each state transition. This puts a fairly low practical ceiling on how fast L1 can go.
L2 uses centralized sequencers to run transactions much faster, but uses a mechanism that runs on L1 to ensure the sequencer can't cheat.
The main mechanisms are 1. optimistic rollups and 2. ZK rollups. The latter, in particular, are fascinating. If you care about distributed systems even a little bit, it pays to suppress your skepticism and learn about how they work.
Good starting point: https://vitalik.ca/general/2021/01/05/rollup.html
Alternatively if you believe this is all just a ponzi scheme involving ape jpegs, bookmark this comment and come back in 3 years.