Side note- you don't want your wire for the cutter glowing red. Bad gases are released by expanded polystyrene foam when it burns.
On to the main event: any engineer worth his salt puts in at a minimum a 50% safety margin. If Radio Shack stamped "2A limit" on your transformer, you can bet that it'd be good to 3A. At least. Possibly more, with forced air cooling. When it fails, it fails by burning through insulation on the wires and shorting out, which causes MORE current to flow which causes more heat which causes...POOF!
To understand the practical limit, let's consider how a transformer works. Imagine an iron picture frame (no glass or anything, just the frame). Take a wire, wrap it around one side of the frame, say, 10,000 times. Now take another wire, go to the opposite side of the frame, and wrap that wire around the frame, say, 1,000 times. Voila! You have a transformer, with a 10:1 ratio. Put 120V on the 10k turn side, you get 12V on the 1k turn side. The flip side is that the actual power has to remain the same, so the equation becomes
Pin = Pout
Vin * Iin = Vout * Iout
So if voltage goes down, current goes up. We'll use your transformer as an example: It's essentially a 5:1 ratio- 120V in, 24V out. It has a 2A limit- so let's consider what the input current is when the output current is at 2 amps:
Vin * Iin = Vout * Iout
120V * Iin = 24V * 2A
Iin = .4A
This is called a step-down transformer. You COULD flip it around, put your 120V line across the 24V terminals, and step up the output voltage to 600V, but I wouldn't recommend it, because the insulation on the transformer is probably not up to blocking 600V, which means you'd have a punch-through of the insulation somewhere, which leads to arcing, current flow, and POOF!
So why the limit? Why can't I just increase the load and draw more and more current from the line voltage side, until I trip my house's breaker? After all, the wall outlet probably has a 15 or 20A breaker, right?
Not that simple. I ignored a little detail earlier- efficiency of the transformer (usually given as the greek letter eta, but frequently shorthanded to a n for people who don't want to find the unicode value for eta. I'll use n). The REAL equation is
Pout = n * Pin
n is usually on the order of 80-90%, or .8 to .9. That means that whatever your input power is, 10-20% of it going to get bled away as heat in your transformer. Let's look at your transformer again:
Pout(max) = 2A * 24V = 48W
Pin = 48W / .85 = 56.5W input power
Ptransformer = 56.5-48 = 8.5W
So, using a fairly unremarkable value of .85 as the efficiency of the transformer, we find that at 2A, your transformer is sucking up 8.5W of power, just to warm itself up. Not too bad, but you'll notice it if you hold the thing in your hand- it'll be quite warm.
In your case, you maxed it out, got WAY too much current flowing in and out of your transformer, and that efficiency killed it. Unlike silicon based electrical elements, something like a transformer can handle quite a lot of current for some time. We just have no way of knowing HOW long...
That 2A rating is undoubtedly for continuous operation, in a somewhat enclosed environment, without cooling. If you put it on a table with a fan blowing across it, that number will go up. Wrap it in a blanket, that number will go down.