RedtailPDX, when you say splice a new wife to it, what does that mean?
Haha, that's just a typo - I meant wire. You're doing it right, cutting the old wire and using crimp connectors (or whatever your favorite method of joining wires is) to splice the wire from the car into the wire from the new stereo. Make sure you tape over any exposed wiring - in the picture you posted it looks like the connector has no insulation and that will eventually arc out and blow your fuse. Just wrap it a couple times in electrical tape.
Okay, so the red wire coming from the car is 12v+ power when the ignition is switched on, and no power when the ignition is off and the key is out right? It was powering the old radio? That should go to the red wire on the new radio.
You also need power from a 12v+ source that is always on (even when the key is not in the ignition) to save your preset radio stations and whatnot. You can pull this from the cigar lighter or the ignition switch or back from the fuse box. This wire goes to the yellow wire on the new radio.
Next you need the speakers - You'll just use the front R & L speakers on the new radio grey and white wires. The old Becker amp should have the speaker outputs labeled L and R; I think white is left, and assuming that's true you would connect them like so:
white plug black wire <-> white/black wire new radio
white plug green & black wire <-> white wire new radio
black plug black wire <-> grey/black wire new radio
black plug green & black wire <-> grey wire new radio
Than you just need to connect the black wire from the new radio to a good clean ground and you should be good to go.
The BLUE & BLUE/WHITE wires from the new radio will be hot when it is on so you should cap them or tape off the ends to be safe. Same goes for the purple & green rear speaker wires you won't be using.