I just bought a new house and of course we had to get a new dog. Well he decided to chew and rip out the thermostat wire from the ac unit. I'm having trouble seeing where the wires are supposed to be connected inside the unit. I work on electronics for a living and this is confusing me. The schematic on the unit is unable to be read. Can some one please help?

Yeah! The control wires and the power feed should both go into a relay inside. If the control wires are a multi wire cable, usually the yellow comes from the thermostat and blue or black is the common from the 24 v transformer.
Any other wires in the control cable are not used at the compressor. In some cases a pair of other colored wire is used in place of the yellow and blue.

I am going to try to upload a picture of my air conditioner with the rigid conduit power feed and the stove pipe over the foam covered tubing and control wires, and the Lab that chewed the foam off, the control wires in 2, and a big hole in the vinyl flex.

Do I match the cables that are already hooked up with the colors of the stat cable or what?

If your cable used the yellow and blue wires, just splice yellow to yellow, and blue to blue. If it used other colors, likewise. If you don't have enough wire left, you can use most any pair of wires of reasonable size. I forget whether regular thermostat wire is 20 or 26 gauge. Since it is floating AC, it doesn't even make much difference which wire goes to which terminal. You just need a loop from the Y contact of the thermostat back to the other side of the transformer.