Because the constitution explicitly grants the president absolute executive power over executive branch (government) of which FCC is part of. If government is a company then president is CEO and can do anything he wants to do.
Of course people can argue about the meaning so ultimately the arbiter of what constitution mean is Supreme Court.
And recently there were several lawsuits in the vain "the president can't do THAT" and while federal judges said "indeed, he can't" and issued injunctions, they were pretty much overturned by higher court or Supreme Court, re-affirming that president does in fact has control of executive branch.
And if you want to game this: if this wasn't true, congress could completely defang the powers of the president by making every government agency (IRS, FBI, FTC etc.) "independent" and de facto giving the power to unelected beaurocrats away from elected president.
And why should you care about this?
Because every 4 years you can vote for a different president.
You can't vote for the head of FCC.