Looking at Coinbase as an example, it's going to work similarly to the Bitcoin Cash hard fork that happened on August 1st.
https://blog.coinbase.com/update-on-bitcoin-cash-8a67a7e8dbdf
Basically, if you don't own the private keys to the wallet you have btc stored in (like if you're keeping them on an exchange such as Coinbase or Bittrex), then you are at the mercy of the exchange and whether or not they will support multiple chains for withdrawal after the Segwit2x hard fork. Again, if it happens in November.
If you
do own the private keys to your wallet, as with a paper wallet or Ledger wallet, then you will have full claim and ownership of the coins on both chains.
With each block that gets mined, the miner can signal whatever type of message they want to broadcast in the coinbase (not to be confused with the exchange company "Coinbase," although this is presumably where they get their name) of the block. You can look at mined blocks
here and
here for example, click to view the scripts and see that BTCC Pool and BitFury are signaling support for the
NYA, or New York Agreement, which was basically a consensus by miners and exchanges to support a doubling of the blocksize from 1MB to 2MB through Segwit2x. This would enable faster confirmation times and lower tx fees, essentially making btc more adoptable on a larger scale.
The only people against this really are the Bitcoin Core developers and Blockstream, who became the predominant entity behind the current Bitcoin client being run on the network. They are essentially arguing that scaling up the blocksize to allow for higher transaction capability and lower fees is a threat to security. They are funded by tens of millions of dollars by VC firms, have argued that high fees to transact with a currency is somehow preferable in the face of basic economics and smalltime user adoption, even
lobbied the SEC to protect the legacy chain from competition. To be fair, it is understandable to an extent that bitcoin maximalists (people who want btc's price to remain high and unthreatened by hard forks and resulting coins that might siphon value away from the legacy coin) would want to avoid hard forks.
Here's a pretty good article about Blockstream.
I don't really understand the argument for keeping the blocksizes small but a lot of powerful people are making it. Hopefully this is more helpful than confusing