You start with some systems already in your ship computer. A number more can be bought from UC in game. Those are all that will show up as explored for you without visiting them.
Systems can have 'discovered by' tags on one or more bodies but they will, outside the systems mentioned above, show up as unexplored to you. You can explore them by scanning the bodies and you can sell that data to UC. If you are the first to sell the data on a body it will acquire a 'discovered by' tag in your commander's name that is visible to everyone else.
Every player has their own list of explored bodies, this is separate to the 'discovered by' list. This list is not necessarily a particularly large volume of data.
The ED galaxy is, mostly, procedurally generated. There are a relative handful of systems that have been hand crafted to conform to what we know about the real galaxy, and to match plot elements, but the vast majority are randomly generated - but random only in the sense that nobody knows what is there without looking, everyone who looks will see the same thing.
So the galaxy consists of 400 billion seeds for the procedural generation, the data for first discoveries just needs to be a list of seeds and planet designations tagged, for each player a list of seeds and planet designations explored. So you go to a system and scan the first three planets then all Frontier need to store against your player in the Explored_Bodies table is System xxxx, Planet 1; System xxxx, Planet 2; System xxxx, Planet 3. It's really not all that much data.
Depending on how their procedural generation algorithm runs they may store the full system details generated for System xxxx somewhere the first time they are generated, but there is a good chance that regenerating every time is faster than dealing with database access so they don't even do that.