"How is it possible to weld with more amps than the amps coming in?"
Because it isn't about amps, it's about WATTS - an older style welder is basically just a glorified TRANSFORMER - and transformers, minus a few percent loss because of efficiency, put out the same WATTS as they take in - if you look at the power lines coming to your house, the lines on the pole aren't welding cable sized EITHER - this is because those lines are typically carrying 49 THOUSAND volts, but only a few HUNDRED amps.That way, those power lines can supply HUNDREDS of houses, each of which might be drawing over 100 AMPS - but those amps are only at 230 VOLTS, not 49 THOUSAND -
IF you do the math, that 49 thousand volt line at (say) 200 amps, is carrying 49,000 TIMES 200 amps, or 9.8 MILLION watts - but that voltage goes thru a transformer (looks like a fair-sized oil barrel, usually up on a pole) - the primary winding of the transformer would be rated at full line voltage, but the transformer steps that voltage DOWN to the 230 volts your house uses. This lets the house draw as much amperage as the power line has TOTAL, but at a MUCH LOWER VOLTAGE.
Your house, for example, even if it's using the full 200 amps, is only doing it at 230 volts - multiply those values together and you get 46,000 WATTS (46 kilowatts) which is NOWHERE NEAR the 9.8 MILLION watts the power line may be carrying, so OTHER houses can still get THEIR power.
Welders - same thing - input volts = 230, amps = 100, means WATTS = 230 x 100 or 23 kilowatts - but welder's secondary winding (depending on weld amps setting) may have open circuit voltage of around 80 volts, which is ONE THIRD of the INPUT voltage - so it can put out THREE TIMES as many amps as it's INPUT because it's not AMPS doing the work, it's WATTS, and watts = amps x volts.
Hope that made some sense, there's actually quite a bit more to it than that (like there's actually another level of distribution by power companies, so your LOCAL power poles usually have more like 10,000 volts instead of 5 TIMES that) - but the PRINCIPLE remains the same... Steve