For Flight #7 slated for 02-Feb-2021, Wikipedia's SpaceX Starship; Testing Program says:
SpaceX attempted to fly SN9 on 28 and 29 of January 2021, but failed to receive permission from the FAA. (164, 165)
Q: Why do Space X starship launches need permission from the US Federal Aviation Administration?
Because it's required by law (51 USC Ch. 509: Commercial Space Launch Activities) and by FAA regulations (14 CFR Chapter III - Commercial Space Launch Activities, Federal Aviation Administration, Department of Transportation) that implement those laws.
Even amateur rockets are subject to some FAA regulations. A rocket going up 10 km is subject to quite a few FAA regulations. Even though SN9 is not going into space (not even close), it has far too much thrust to qualify it as an amateur rocket. The launch of SpaceX's SN9 is categorized as a commercial space launch, and FAA approval is required for that.
Update
My answer does not address the issue of "why?" Shortly after aviation started more than a century ago, incidents and even fatalities became ever more frequent occurrences. The US government passed multiple laws, starting with the Air Mail Act of 1925 to attempt to make air travel safer, more common, and more lucrative. And it has worked. US commercial aviation makes about 5670 flights per day, and incidents are extremely rare.
The mindset of the modern Federal Aviation Administration (FAA) goes a bit against the mindset of rocketry. Suppose that one out of every thousand commercial airplane flights results in death to all. Given the current flight rate, that would mean six fatal commercial airplane crashes per day in the US alone. Hardly anybody would fly in a commercial airliner were that the case. Modern aviation is extremely safe. It is arguably the safest mode of transportation. Suppose even one out of every million commercial flights resulted in multiple fatalities. That would given the modern FAA qualms. With over two million commercial airline flights per year in the US alone, that would mean two major incidents per year, in the US alone.
An incident rate of one out of every thousand rocket launches remains a mere pipe dream. That means that the mentality of the modern FAA, where a one out of a million incident rate would be a bad sign, goes a bit against the grain of current rocket launch statistics. Nonetheless, the US Congress has mandated that the Department of Transportation (and hence the FAA) foster and regulate the commercial space industry.
To expand a bit on David Hammen's answer, the reasons for the regulations requiring FAA permits for rocket launches are related to public safety (or sometimes the egos of bureaucrats, but mostly public safety.) There are a couple of particular areas that these fall under:
Obviously, rockets carry a lot of fuel and often very toxic materials (e.g. hydrazine or sometimes even radioactive material for radioisotope thermoelectric generators.) Rocket launches don't always go as planned, which sometimes leads to large explosions and/or highly toxic stuff landing downrange (or just exploding on the launch site.) Thus, making sure that the risk of such an incident harming people and property is minimized is important. The FAA is the U.S. agency in charge of safety for stuff that flies, so this falls under their purview.
Spacecraft tend not to be the only vehicles flying around in the atmosphere. In general, they must share the atmosphere with aircraft, even if only very briefly. When you're flying around in your Cessna (or your Boeing, for that matter,) it's not exactly easy to see and avoid something that suddenly lights up underneath you and accelerates quickly to supersonic speeds straight up in your general direction. Thus, just as aircraft have regulations to prevent them from flying into each other, so, too, do spacecraft have regulations to prevent them from flying into aircraft.
In particular, when a rocket launch is scheduled, a Temporary Flight Restriction (TFR) will be issued telling aircraft pilots to stay away from a set boundary of airspace through which the rocket is expected to transit the atmosphere for a set period of time. A Notice to Airmen (NOTAM) will be issued to advise pilots of the TFR. Checking the current NOTAMs is part of normal preflight procedures for pilots, so this ensures that they'll be aware of the launch (assuming they actually conduct a proper preflight briefing.)
Of course, the FAA is the agency in charge of maintaining aviation safety, so this concern also falls under their area of regulatory authority (and they are the ones who issue the TFRs and NOTAMs.)
In this case, this part of the launch licensure requirement is actually useful to the organization launching the rocket. By getting a temporary flight restriction put in place around the launch site for the desired timeframe, they can ensure that some random guy buzzing around in his 172 isn't going to stray too close to launch site during the launch window. Launch windows are often very short (sometimes even instantaneous,) which means a launch might have to be scrubbed entirely if an airplane wondered too close to the launch site at the wrong time. Having a TFR in effect means that both pilots and air traffic controllers will be aware of the launch in advance and will plan for air traffic to avoid the area during the active time of the TFR, including the launch window.
Everything that goes above FL 180, about 18'000 feet needs a flight plan, which has to be filed with the FAA that can approve or not
