That black wire, looped around one switch screw and continuing to the other switch (conveniently highlighted yellow in your last picture) is the "hot" power supply coming into the switch box and eventually makes its way back to your breaker box.
You can confirm this by carefully measuring with a multimeter when the power is on. Touching one probe to that black wire, and the other to the big bundle of white wires (with the red wire nut removed) should give you ±120v no matter what position either of the two switches is in.
Once you've confirmed this (or taken my word for it), TURN THE BREAKER OFF AGAIN. Then you'll want to disconnect the wire from that screw and cut it. It looks like you've got plenty of wire in the box (which is good!), so cut off all the exposed wire to get rid of the bent part. Strip back each of the newly cut ends to the length specified in your instructions (usually about 1/2").
You'll also need to remove the "backstab" wire (marked green in your overlay pic) from the back of the switch. Backstabs are notorious for failing over time, causing arcs and, potentially, house fires. Plus, you need to get this switch out of the box, so all the wires have to come off it. To remove that wire, you'll push a small screwdriver into the little slot next to the backstab hole and pull the wire out. You may have to twist & wiggle the wire a bit to get it out. Cut it if necessary.
You now have three wires to work with - 1 "green" and 2 "yellow" (they all have black insulation and all serve as "hot" wires, I'm just going with your overlay colors for ID purposes). You'll need a new wire nut of the same size as the red one that's already there. You'll add 1 of the black wires from your switch to that bundle (total of 4 wires) and tighten the wire nut down on the whole thing. Once you think you've got it good and tight, hold the nut then try to pull each wire out. This will ensure that they're all tightly held in place. If any come out, you had a loose connection that could have arced and caused a fire, so you need to try it again.
From there, continue to install per the instructions that came with your switch:
- Remove the other backstabbed black wire (light blue) from that switch and connect it to the other black wire on your new switch.
- Connect the green from the new switch to the bare ground wire bundle
- Connect the white from the new switch to the white neutral wire bundle.
As noted in a comment, it may be that the hot power coming into the box could be the "orange" wire.
If that's the case:
- Remove the "orange" wire from the right switch
- Clip the "yellow" wire as directed above
- Put both "yellow", the "green" and the "orange" under one wire nut with one of the black from the new switch
- Put the now loose end of "yellow" back on the right switch (use the screw to properly fasten it, don't use the back stab).
Why is it that the left switch has two hots ("yellow" and "green") leading to it?
It doesn't. The "green" is an unswitched hot leading off somewhere else, possibly an outlet or another switch somewhere.
