Since I don't think I saw this question addressed directly: the "A" settings on your meter are there to measure current (i.e. Amps), never voltage.
Yeah that I know, the problem was I had it in voltage mode and tried to directly measure AC voltage through that port.
Yes, but you cannot measure voltage (or the other non-current modes) through the current jack. Regardless of whether you INTENDED to measure voltage, your meter was configured to measure current despite what position the dial was in. Some meters attempt to protect you from a mismatch between the measurement range selected and the jacks being used, by either physical means (shutters that close the amp jacks in voltage modes, and vice-versa, and physically block the dial from selecting an incompatible range) or electronic means (such as a loud beep, flashing lights, or a warning message on the display). Unfortunately, the majority of meters, especially the low-cost ones, will not prevent you from making this mistake. The only safety system you can truly rely on is the one between your ears.
Voltage measurements do not use the fuses.
Wait, they don't? So the fuse was bypassed when I was using the 10A port to measure voltage?
No, the 10A fuse is ONLY used for the 10A jack. You cannot measure voltage through the 10A jack, it is for measuring current. The voltage jack bypasses all fuses.
As we're explaining, the current jack places only a low impedance shunt (essentially, a short piece of wire) and a fuse between the probes. That's it. So using the probes in parallel with a circuit, as you would when measuring voltage, is shorting the circuit and allowing maximum current to flow. Which blows the fuse (or minus a fuse, blows up something in the meter if not the meter itself). Again, to measure current you must break the circuit and put the probes in series with the circuit, not in parallel like for a voltage measurement. Or else use the aforementioned current clamp, which is by far easier if accuracy isn't critical.
The voltage jack has a high impedance and does not need a fuse to limit current.
Placing probes from live to neutral when measuring current, as you would if measuring voltage, is directly shorting them together with only the very low impedance current shunt as a load. Pretty much the same as if you stuck a fork in an electrical outlet! Fortunately Klein are not bottom barrel meters so its damage is likely limited to the blown 10A fuse. Cheaper meters could potentially break down even more catastrophically (though unlikely at only 120V mains, as the energy isn't all that high compared to 230V or 480V or higher systems). If it weren't fused at all, the meter would be damaged probably beyond (practical) repair.
So it did go through the fuse then?
Yes. That's why the fuse is blown.
No offense intended, but I strongly suggest you look up how a multimeter works, because this is basic DMM operational knowledge and without it, you might not only destroy equipment, but (at least with an unsafe meter) potentially hurt yourself, burn down your house, etc.
I used to, it's just that it has been a while.
I understand. But this is fundamental knowledge and based on the conversation here, you're still figuring it out. I think all of this will become clear to you if you spend some time on some basic education in DMM use. There are countless articles and YouTube videos on the subject. Some basic meters (like the Fluke 101, still a Fluke but inexpensive and made for simple use) don't even have a current jack, as honestly for most homeowner use the need to measure current is an uncommon one compared to voltage. They are the safest to use because you can't make this mistake with them.
And to be clear, I'd say virtually ALL of us, even seasoned, professional engineers and electricians, have made this mistake at least once in our lives. However it isn't typically due to a lack of understanding of how the meter works, but rather due to carelessness in not paying attention to where the probes are plugged in as you're switching between different measuring modes. You switch the dial but forget to change the probe jacks, or vice-versa. So we're not judging you for it happening, we're just admonishing you to learn a bit more about your tool and what you're trying to do with it, as it's potentially very risky to your person and property to not have a fundamental understanding of how DMMs should be used.
As to the damage to the probe, it looks minimal and entirely salvageable. Even if it weren't, probes are not expensive to replace. And you don't HAVE to buy replacement Klein probes. You can buy entirely serviceable probes for less than $10.