For running that spindle, yes, you'll need an inverter, also called a Variable Frequency Drive. It produces 3-phase power from the single-phase power you've likely got, and feeds it to the motor at varying frequencies that determine its speed (but in a narrow range; these motors can't run slowly). You can use a regular hand router, but they're noisy and don't last long in this application; 3-phase motors are more powerful for their size, quieter, and just keep going.
For those motors, I'd think a higher-voltage power supply would be better. The motors want 51 volts to run optimally, and the drivers say they'll take 50v, so a 48v power supply would probably be safe.
If you're cutting wood, save some money in your budget for a good dust collection system; these things make prodigious amounts of dust.