Need understanding on amperage

the 60 amps is stated under "current capacity" under the specifications section on their website.

Like a few have already stated, power is a better way to look at the system because power is constant even when current and voltage are changing. What that means is that the amplifier has transformers to reduce the voltage levels. So current out of your amp can be higher than the current at the wall since the voltage is lower. Your amplifier spec is stating 60 amps to either show a short term power storage peak output or else to say that the transisitors are robust enough to withstand that current draw for some period of time without popping.
A 120 VAC wall plug can safely handle about 1200 Watts. Looking at high power home appliances, I doubt you could find a microwave oven or toaster rated higher than that. I searched far and wide for a 2-slot toaster rated at 1200 Watts so I can have decent toast. Anyway, 1200 Watts is a 10 amp load and anyone with an older home will tell you, don't try to run a microwave oven and a toaster on the same circuit at the same time. So like someone already said, if you were to input 60 amps into a 1 ohm speaker load, that would be 3600 Watts. It may be possible for some milliseconds with the help of the energy storage capacity in your amp, but not continuously simply because a house circuit could not handle that load.
My amp pulls about 5 amps all the time out of the power conditioner. I have seen it peak at 5.7 amps if I have the music cranked. My voltage hangs in there at around 122 VAC because I have a dedicated circuit. So I am pulling up to 700 Watts total- but only for a second or two. The amp output would be something less than 350 Watts per channel. Then my wife tells me to turn it down and my current draw is back to 5 amps. If my speakers have say a 4 ohm average load, then the current draw into each speaker is no more than 9 amps. That is 18 amps total, even though the current draw at wall is only 5.7 amps. These are rough examples because the power output to the speakers will be a little less than the power into the amp due to losses through the amp (in the form of heat).

Seriously, is anyone suggesting that the output section of this amplifier can deliver 3600 watts even briefly?? The output devices would be destroyed. Now if you assume a 4 ohm load rather than 1 ohm, the power is then 14,400 watts. Obviously this 60 amp figure has to do with something else- the math isn't lying...

This rating has nothing to do with the output power of the amp. It does get conflated that way quite often though and I think a lot of manufacturers like it that way.

Pay attention to what Ralph from Atmasphere is saying in this thread and you will be for the better.

Seriously? You better believe it. Real life sounds have sharp, high transient peaks that seem like no big deal but require massive power to faithfully duplicate in a sound reproduction system. The amplifier has to generate sufficient power to overcome the mechanical and electrical inertia of the speaker drivers to make the slap of a drum or pluck of a string sound real. The more power on tap in the amp, the more realistic the sound. That's why amplifiers have big transformers and capacitors, so they can hit those momentary high Wattage peaks if just for 10 ms or so. Not 14,000 Watts probably, but 3000 Watts is possible for a very short time. That power is needed to make the fast rise times so the music doesn't sound blurred or distorted. That's the difference between listening to a hifi system and a clock radio. For sure, the transistors will pop like fuses if they have high current levels for too long. It is all about heat buildup inside the semi-conductors. His amp manufacturer rates the transistors for 60 amps, but no idea if that is for 1ms or 1 hour. The circuits may be capable of those power levels, as I am thinking for robustness, but agreed the amp could never generate those power levels for more than milliseconds (and likely not 60 amps) being supplied by a standard 120VAC wall plug.
One example that comes to mind is the 1812 Overture. I have never heard the cannons faithfully reproduced by a stereo system. The power to do it would be at the industrial level, as well as the speakers to handle that sound pressure level. I mean you know they are cannons that are being fired, but not even close to the impact and shear power of the real cannon shot. I think it would be fun to play the record and have someone fire real cannons at the appropriate moments, but the neighbors might not think it so much fun.