The rating of a router is peak horsepower, not mean or average horsepower. As one is routing, the router is generally taking much lower power than the rated peak power from the receptacle.
As one routes, the router might hit a tough spot, say a knot in the wood. The router struggles to keep going, drawing more power from the receptacle. Once through the tough spot, the router finds things easier. So, the power drops again. This peak in the power drawn by the router should be not more than the rated peak horsepower.
Okay, one asks why the circuit protector (fuse or circuit breaker) doesn't trip. Well, that has to do with the response of the protector. A 15 Amp protector will carry 15 Amps, or less, indefinitely. It may carry 30 Amp for say 10 seconds without tripping. 60 Amps for 2.5 seconds. And so forth. (An example curve showing this relation is at
http://downloads.eatoncanada.ca/dow...ircuit Breaker Time-Current Curves For QUICKLAG Industrial Circuit Breakers.pdf)
Back to the question: 3 HP would be about 2241W or 18.7 Amps at 120V. If it takes one three or four seconds to get through the rough spot, then the 15-Amp protector is not likely to trip. If one keeps the router plowing through a rough spot for say two minutes, without letting up on the router, then the protector is likely to trip.
Ah ha! Not magic; just smoke and mirrors.
Cassandra