Anyone know why there is a jumper between the lugs in my sub panel?

That jumper is there because whoever wired your garage only ran a single 120V hot leg to the sub-panel. In a standard setup, you have two seprate bus bars fed by two different 120V legs to give you a total of 240V. By jumping the two main lugs together, they've made it so every breaker slot in the panel is powered by that same single leg. This is a common shortcut when someone doesn't want to run a proper 4-wire feed, but it limits you significantly.

You won't be able to run any 240V appliances like a large compressor, a welder, or a car charger with this setup. Because both sides of the panel are on the same phase, a double-pole breaker will just give you zero volts across the two hot terminals. It also means you're potentially overloading the neutral wire if you aren't careful, though in a small garage sub-panel, the main concern is usally just the total amperage draw. That loose red wire is a massive red flag. It's likely the second hot leg of the cable that was never connected at one end or the other. If it’s live and touches the casing, you’ve got a short circuit or a hot enclosure. You need to check it with a **non-contact voltage tester** imediately. If it's dead, it still needs to be capped with a wire nut. Leaving a bare, stripped wire hanging in a panel is bush-league and dangerous.

Check the gauge of the wires coming in. If they look thin, like 10-gauge wire, you're only looking at a 30-amp service for the whole building. If you try to run a table saw while a space heater or a vacuum is going, you're going to be walking back to the main house to flip the breaker constantly. Honestly, if you're seeing jumpers and loose hots, I'd bet there are other issues like missing ground rods or bonded neutrals that shouldn't be. Getting a pro to at least look at it and verify the feed capacity is a smart move.

That neutral overload part has me sweating a bit. I just checked the main house and that red wire is acutally connected to a double-pole breaker on that side! If I pull the jumper out and hook that red wire to the second lug instead, does that fix the whole 'single phase' problem? Or do I need to worry about the white wire being too small to handle both legs now? I'm trying to learn the 'why' behind the code here before I call an electrician and look like a total idiot.

If that red wire is alredy landed on a double-pole breaker at the house, you're halfway there. Pull that jumper out and land the red wire on the second lug—that'll give you your 240V spread. Regarding the neutral, it’s actually the opposite of what you’re thinking. In a sub-panel feed like this, the neutral only carries the imbalance. If you have 15 amps on one leg and 10 on the other, the neutral only sees 5 amps. It’s way safer than the current jumper setup where everything piles onto the neutral. Just double-check that your ground and neutral bars aren't bonded together in that garage panel; they should be isolated from each other.

Since the previous owner was sketchy enugh to leave a hot wire dangling, you relly need to check if the neutral and ground are bonded. In a sub-panel, they have to be separate. Look for a green screw or a metal strap connecting the neutral bar to the panel casing—if it's there, pull it out. Also, verify the breaker size at the main house matches the wire gauge. If they used 10-gauge wire but slapped it on a 50-amp breaker to stop it from tripping, you're asking for a fire. If everthing matches up and you land that red wire, you'll finally have a proper setup.

While you're in there landing that red wire, check the torque on those lugs. If the previous guy was lazy enugh to leave a hot wire flapping in the wind, I gaurantee he didn't use a torque screwdriver to set those terminals properly. Loose connections create heat, and heat leads to arcing or melted insulation. Also, check the bus bars for any signs of pitting or discoloration where that jumper was sitting. If it wasn't seated perfectly, you might have some oxidation that'll mess with your voltage once you put a real load on it. Give everythign a good tug test before you flip the main back on.