That Buffalo router comes w/ an integrated WAP (wireless access point). When you switch it to AP mode that just disables all the routing features leaving it as a simple WAP. But a WAP is something you connect TO not something you connect FROM. When connecting it to your XBOX you want it to act as something you connect FROM over TO your existing wireless router.
The WDS bridging feature does offer the ability to connect FROM that Buffalo TO another WAP. But there's a catch. WDS is not a certified wifi protocol so implementations vary. What that means to you is the very distinct possibility that it won't work unless your existing wireless router a) supports WDS and b) is also a Buffalo wireless router.
So minimally you need to determine if your existing wireless router supports WDS. If not you're dead in the water. If it does then it may work or it may not due to incompatible implementations of WDS. But sometimes you get lucky and it works even across brands (but usually not). And even then some implementations of WDS have other limitations (e.g. WEP only).
That's why the better choice would have been a wireless Ethernet bridge. Without that capability you're left w/ WDS and that's just a PITA most of the time.
So before you take another step tell me the make & model of your existing wireless router so we can determine if this is even remotely possible.