How do I replace a 3-way switch that is connected directly to the house wiring?
I presume your new 3-way dimmer switch has 4 braided wires coming out of it. Green (ground), a couple of red wires (travelers) and a black one (common/hot).

Yes, just cut the wires from the old switch (leave them as long as you can) and connect them to the wires on the dimmer switch using small wire nuts (no bigger than the standard yellow ones; the dimmer switch probably came with some wire nuts). Make sure you can't pull any of the wires out of the wire nuts, and make sure the stripped ends of the wires aren't so long that you have bare wire outside the wire nuts. You can wrap the wire nuts with electrical tape for good measure.

I presume you know that of the three conductors (plus the ground wire, if there is one) connected to the three way switch, only 2 are travelers and the other one is the "common." The common, depending on which switch you're looking at, is either supplying power into the 3-way switches, or feeding power out of the switches into your light fixture.

The travelers can be hooked up in any order, but the common must be connected to the common terminal/wire on the switch, or it won't work.