I saw on wiki that a boeing 737 flys at about 38,000ft and a Concorde flys at 60,000ft. What made it possible, and advantageous, for Concorde to fly so much higher?
It was necessary for Concorde to fly much higher, and not a voluntary decision. A higher cruise altitude increases the demands on the airframe (examples: Combustor size, pressurization) and is only accepted when it comes with sufficient benefits.
An airplane needs a certain wing area for realistic take-off and landing distances. Also, details like the maximum tire speed need to be observed, so the wing area must be large enough to create enough lift at low speed.
Lift is proportional to the square of airspeed and air density. The faster a plane flies, the lower the density can be to create sufficient lift. By flying more than twice as fast as a Boeing 737, the Concorde could fly at a fourth of the density by this reasoning. Air density (symbolized by the Greek letter $\rho$) halves approximately with every 5500 m of altitude increase.
However, the slender wing needed for low supersonic drag is not as efficient as the sub- / transsonic wing of a 737, so a little more density was needed for the Concorde. The 737 is designed for cruise between 8000 m ($\rho$ = 0.526 kg/m³) and 12000 m ($\rho$ = 0.312 kg/m³), and the Concorde flew between 16500 m ($\rho$ = 0.154 kg/m³) and 18000 m ($\rho$ = 0.121 kg/m³).
The air is much thinner up that high and therefore there is much less drag/resistance, therefore an airplane won't heat up as much as it would going the same speed at a lower altitude, and it requires much less thrust/fuel as well (economics matter too when you have paying passengers).