Most likely they mean they will give you a point to point, layer 2 connection. So for example, if you have 192.168.1.0/24 on your current site, they will give you a layer 2 link to your remote site, and your remote site will be on the same 192.168.1.0/24 block.
So if your switch at HQ is 192.168.1.1/24, you could plug the carrier's "router" device into your switch there. You could put a switch at the remote site, hook it to the other "router" from your provider, set it up as 192.168.1.2/24, and ping 192.168.1.1 because they will be on the same subnet. Devices on both the HQ site and remote site could use the same default gateway to get to the internet.
So essentially, both of your sites will be on the same exact network. A straight layer 2 connection between the two sites.