How should I connect a replacement dimmer switch?
There are 2 things I see from your photo and video. 

First I believe the two black wires in the box are a source wire (120V) and a continuation of the source to another box. This means with both of them apart, something else in your dwelling is not working. So check that out. The red wire is probably the switch leg to your light or lighting fixtures. 

Second it looks like the replacement dimmer you have is for a 3-way dimming set up. The 2 red wires are the carriers and the black goes to the light fixture. 

If what I said is correct then you can connect the two black conductors in the box with a one red conductor on the dimmer and wire nut off the second red wire. Then you can connect the red conductor from the box to the black wire at the dimmer and that should work. You might just take it back and get the proper dimmer switch.

WARNING I can't be sure. Since I am not on site and cannot test the conductors in the box and cannot read the information on the dimmer. You might check the box the dimmer came in there is usually schematics on connecting the dimmer on the inside of the box. Also take a meter and identify the wires inside the wall junction box.

Good luck and be safe. Make sure your conductors are all turned of at the panel before trying any connections.