Most ISPs have to buy service from other ISPs to be able to reach every part of the internet. That is called "transit". If you start a small ISP you usually buy transit from one or more bigger ISPs. They will have connectivity to the whole internet, and they will sell that connectivity to you.
You can also set up direct connections to other networks. This is called "peering". Usually those are the networks you exchange a lot of traffic with. Paying a transit provider for the packets you exchange can be a waste of money if you can easily connect directly to the other network and cut out the middle man. Sometimes other networks want you to pay them to set up a direct connection, but in many cases the peering if settlement-free.
The problem with setting up direct connections is the number of network interfaces you need, pulling cables between routers etc. Because of that there are "Internet Exchanges" (IX). An IX is basically a large switch (large IXes are more complex, but from the outside they function as a large switch) that many internet providers and enterprises connect to. That way they can all talk to each other, if they want to. One connection to an IX can let you set up direct peering with hundreds of networks.
As you can see there is a reason you pay your ISP. They invest in routers, connections, and very likely pay larger ISPs to transport part of their traffic. You could set all of that up for yourself, but be prepared to invest tens of thousands of dollars/euros/etc for equipment and connections. And then you'll probably pay more than you pay your current ISP for the transit costs, as you won't be able to negotiate lower prices for buying in bulk.