Back then you bought them on a site that connected buyers with sellers. Now that I am coming to the topic again, I see things are diffetent. I don't see any p2p exchange, you buy and sell them from big brokers.
Really? I was pretty sure that most sites are just connecting buyers with sellers - GDAX, Bitstamp, Kraken and the majority of others appear to handle their customers' coins in this way. For example, on Bitstamp's FAQ, they point out that they don't actually sell coins themselves.
As for the more direct "connecting buyers with sellers", LocalBitcoins is still very popular for a variety of different payment methods.
Now, where do they get them from? From miners I guess. So how do you buy bitcoins from miners? How do miners sell them?
Miners sell them similarly to how other users do - however is most convenient for them. That means usually on exchanges or OTC if necessary.