Writing all order transactions to the blockchain gives the user full transparency to what is bought and sold. Implications of this are difficult to understand as this is a very abstract thought.
A simple example might give an idea of how important this is.
The biggest reason we have governmental financial oversight is that we have to trust what people say they do with our money. I give you $100, I need to trust that you put those $100 into an investment of my choosing. I, as a person, cannot really be sure that this happened. Yes, the database says that $100 went in and most likely(almost 100% certain) I will get my $100 if I ask for it. But there is always a tiny little chance that somebody could be doing something wrong/illegal and therefore we have government institutions going into financial companies to check if all is ok.
Having transparency, 100% removes this uncertainty, at any time. I, as a person, can see where my money went and what happens to it. This cannot be faked. Getting the information depends on how advanced you are, this might simply be a website you open, that reads the blockchain and displays this information. If you are more advanced maybe you have it in Excel, Power BI, or other analytics tools. Maybe you are a full geek and read it from a blockchain explorer such as etherscan.
You can always trace who executed an order and this cannot be changed. As long as your wallet is safe, nobody can do anything in your name.
The user is authorized using his wallet. To have access to the system, the user logs in with his wallet by signing a message. This gives cryptographic certainty that the user is who he says he is.
Before any order can be made in the system, the user needs to authenticate using KYC or another authentication process acceptable by the market that will be traded in.