I work for a CDN, but am not a networking expert. I will try to answer as best I can:
Yes, it is a physical location. Often called 'an exchange', there is usually a building in major cities that most networks connect together at. You rent from the building owner, and they provide a 'drop' which is basically a cable you can connect to your core router in your part of the datacenter you are renting (your 'rack' or 'cage')
From that point, you set up BGP sessions with your 'peers', other networks you have agreements with. Sometimes those agreements are for 'transit' (which means you pay per bit sent across the wire, but you can send all your traffic that way) or they are peers you don't pay (you might have other agreements, like we only send this much traffic or that hte inbound and outbound traffic has to match)
According to that peering link from the above poster, these guys will peer with anyone. They basically want as many peers as possible, and are hoping those peers agree to route as much of the internet as possible.
The way BGP works is that each side of the BGP session tells the other 'i will accept and route traffic for these subnets', for example "i will accept packets destined to any IP address in the 1.2.3.240/24 space" The /24 is what is known as a CIDR, and is a way of writing "all the ips between 1.2.3.0 and 1.2.3.255", 256 IP addresses.
So if the core router on my side gets a packet from my network destined to 1.2.3.123, I could send that packet on to the peer who is advertising that they will accept packets destined to 1.2.3.240/24. You need to have a peer for every IP address to be fully connected to the internet. If you don't, you would normally pay for transit, which will route for all IPs.