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This is a common question. The answer is that routers use their "own IP address". In many cases, that's a 10.x.x.x (or 192.168.x.x) address local to their network. Indeed, it's not necessary for routers to have a public Internet address at all.
In other words, the question you should be asking why routers have public, routable IP addresses in the first place, instead of always using addresses that are only locally meaningful, like 10.x.x.x.
Remember that the source and destination IP addresses of a packet the router is routing are on the far ends of the network. Nowhere in the packet does it contain the router's IP address. QED: the router doesn't need an IP address.
The reason traceroute almost always shows Internet routable addresses is pretty much just because of traceroute -- admins put public routable addresses in their routers so traceroute will work.
