How urgently do I need to replace my tire? When I came home today from by bike ride I found my tire like this:

I understand that I'll need to change my tire. My question is how urgent is this? Can I just keep using the tire until it stops holding air, or is this dangerous, and the it might pop when I'm on the bike?
That’s really bad. Replace that tire immediately, before the next ride for sure.
When you can see the cloth skeleton of the tire, that means it’s finished. The cloth is nowhere near as abrasion resistant as the rubber, and you’ll soon rub through it. When that happens, the tire is likely to fail dramatically in something known as a “blowout”. This is a very sudden event that leads to a high risk of crashing. Don’t tell yourself that you’re skilled enough to ride it out, it’s just not worth the risk.
Furthermore, see how the side of the tire is fuzzy? That means the cloth on the sides is damaged. If it tears, again, your tire might explode and you will likely crash. Do not try to risk it.
Edit: Realize that it is not the tire which holds air. The tire itself is porous. From a pneumatics viewpoint, it only serves as the mechanical device resisting the 70+ psi outwards force from the air. Inside the tire, there is another rubber donut, which is an impermeable bladder known as your “inner tube”. This inner tube is not strong enough to hold that amount of pressure by itself. Henceforth, when you wear through enough of the outer tire, it will fail to be strong enough to resist that 70psi and subsequently tear. The weak inner tube will then expand through the tear and burst because it isn’t being contained by the tire anymore. Although it will become easier to puncture the tube as the tire wears down, it will not give you any major advance warning (you might hear cloth ripping and threads breaking, or you might not. Depends on your luck and the exact cause of tire failure. Hitting something like a sharp rock definitely will not give any warning) before it lets go altogether. Since it is only the inner tube which holds air, the tire will keep on going until it instantly rips apart.
Congratulations for riding so much you’ve worn a tire out though! That’s awesome!