But after all the blocks are mined, how does the blockchain even work?

Your confusion is why I hate the term "mining". Call it "transaction notary service" and things make more sense. Currently the people with hardware get paid by a combination of transaction fees and inflation of the bitcoin supply. Once all blocks are "mined" they get paid for being notaries only with the fees.

Every time a block is mined, miners are paid in transaction fees + newly generated coins. After 21 million coins have been generated, miners will only be rewarded transaction fees => it doesn't mean blocks will stop being mined; blocks will keep being mined, but without generating new coins out of nowhere.

How is the amount of a transaction fee determined?

It’s offered by those wishing to include their transaction in the next block. Miners look at all the pending transactions and select the most lucrative ones.

Every block has a limited amount of space for transactions (1 MB previously, 4 and up to 8 technically with segregated witness?) , the person mining the block includes transactions based on fees provided by users initiating the transaction.

If there are tons of people looking to make transactions, fees go up or down based on people's willingness to pay to be included in the next block.

In practice Segwit only increases the blocksize to 1.4 MB for normal transactions and theoretically to 4 MB, but then blocks are filled with special kinds of transactions people don't really use.

It also doesn't really increase the blocksize, but move some data outside of the blocksize calculation to make room for more transactions. The important difference is that Segwit is opt-in and depend on usage for it's effect, while a blocksize increase would immediately increase transaction throughput to its full capacity.

Bitcoin Cash chain has done increasing the blocksize rightfully.

Essentially by market forces. Transactions with too-low fees don’t get added to the chain.

Other people have answered the question quite well, but I just want to point out the Bitcoin FAQ [1], which is pretty well written. It is good to read the entire faq to get an understanding of not just bitcoin, but blockchain in general.

[1] https://bitcoin.org/en/faq#mining

In addition to the other answers, there are some cryptocurrencies where the block reward never goes to zero. Look up how it works in Monero for example.

There are also cryptocurrencies where the block reward is fixed forever, so that miners at launch are rewarded no more than miners decades later. Note that such a linear emission is still disinflationary in the sense that the yearly inflation rate tends toward zero.

Transaction fees.

Blocks continue to be mined. It is just that the reward for doing so becomes zero

This is wrong and doesn’t even make sense. If there were no reward, blocks would not keep getting mined.

It's kinda wrong in the sense that the fees still exist, but it's right in the sense that the "block reward" is terminology used to refer to the fraction of miner revenue in each block which increases the total supply. That part does indeed stop after 21m according to the current codebase.