Chances are that there is a "plate" (a horizontal piece of wood) at the bottom of the wall the wires are in. You need to get a flexible drill bit (such as these) to drill a hole through the plate and into the floor below.

Hopefully that comes out in a wall that's directly below the one on the upper floor. Then you can just cut a new hole in that wall to mount an electrical box where you can terminate the speaker leads. (Since it's low-voltage, you can use a low-voltage "box", which is really just a shallow backless frame to mount a wall plate to.)

If it's not above a wall, it should come out in the joist space and then you'll have to run it to a wall. (If it's an unfinished basement without a ceiling, you're in luck. If not, that's a lot of dealing with ceiling drywall or plaster.)

To pull the wires, you could get a fish tape, but it's probably overkill. You can just tie a string onto the end of the drill bit, pull it back through the hole, then tie wires to the end of the string in the basement, then pull them through the hole. Leave string with the wires in case you ever need to pull anything else.

So, assuming you have a drill and some string, the only additional thing you'd need to get is the drill bit.