A microwave powered drone is totally different than a microwave powered rocket. The experiment with the drone uses power from the microwave beam to generate electricity to turn propellers. So in theory a microwave powered mother aircraft could lift a rocket to some altitude but with propellers the limit is probably about 20,000 ft due to the thinness of the air at those altitudes. Conventional jet powered can lift a rocket to 50,000ft before releasing the rocket so it's far easier to do that with conventional jet powered aircraft and doubtless cheaper. Once you've reached those sorts of altitudes you then need a rocket which is expelling propellants to produce thrust in accordance with Newton's Third Law to lift you up to orbit, propellers are useless in the vacuum of space. So whilst in theory microwaves could beam power to a rocket you still need propellants on board to be expelled. I suppose you could use the microwave power to heat hydrogen gas on the rocket and expel it (Hydrogen would be a good choice due to the low atomic mass and consequent high specific impulse).
Moreover given the amount of power required beaming hundreds of MW of microwave power into the sky would have some environmental and safety implications, I think any geese flying through the beam would fall out of the sky ready cooked!
Microwaves have been used to transmit power since the late 1970's.
In 1979, NASA powered a 100 W light bulb on the Big Island of Hawaii from a radio transmitter located on an island a bit over 75 miles away. this was done as a part of a feasibility study on the technologies for Solar Power Satellites (SPS). Powering a drone or a small rocket is not really any different. There have been several radio controlled small aircraft that were microwave powered since then.
For deep space the problems are mostly with keeping the beam focused on the target. Efficiency is generally around 25%, meaning you have to send around four times more power than the vehicle uses. Not really a problem in space where you have twice the solar intensity and where size is whatever you need within the limits of tidal forces.
The US Military has some plans for tests on powering remote military bases using such antennas.
The receiving antenna is technically a rectenna, that is an antenna connecting to the power lines through a rectifying diode with a capacitor. This converts the AC of the HF radio signal to DC for power usage. In the 1970's the best DC to AC conversion for standard AC power was using a motor generator. Today we would use silicon devices such as an inverter for this task.
Plans back in the '70's were for square mile antenna arrays that would provide gigawatt power levels for months on end. The antennas would have been a lightweight space frame with the mounted antennas on it, and farmland underneath.
On a spacecraft, you would have a large rectenna connecting to the power bus. The rocket, most likely an ion or plasma engine, would provide a small push for a long time while using very little fuel.
I don't believe though that such an engine will ever provide enough thrust to launch from the surface of the Earth. The Chinese are working on a plasma jet engine that might provide that level of thrust. That's something to look for in about fifty years.
But these plasma or ion engines still have to carry along some fluid for thrust. They do give high values of Isp.