What is Freedom of the Press?

According to Donald Trump, ""It is not 'freedom of the press' when newspapers and others are allowed to say and write whatever they want even if it is completely false!"

That is precisely what freedom of the press says. The Founding Fathers knew what they were doing and saying when they included this in the First Amendment. 

“If men are to be precluded from offering their sentiments on a matter, which may involve the most serious and alarming consequences that can invite the consideration of mankind, reason is of no use to us; the freedom of speech may be taken away, and dumb and silent we may be led, like sheep, to the slaughter.”
— George Washington

“If there is any principle of the Constitution that more imperatively calls for attachment than any other it is the principle of free thought — not free thought for those who agree with us but freedom for the thought that we hate.”
— Oliver Wendell Holmes Jr.

From the First Amendment Center: "Unless restricted by a valid prior restraint (which is rare), the news media are free to publish any information or opinion they desire. This freedom, however, does not immunize them from liability for what they publish. A newspaper that publishes false information about a person, for example, can be sued for libel. A television station similarly can be sued if it broadcasts a story that unlawfully invades a person’s privacy. Because such liability can be staggering, most journalists strive to exercise their freedom to publish in a responsible and ethical manner."
and

"The reporters and editors at my local newspaper are highly biased. Is there a federal regulator who should know about this?

No. Under the First Amendment, newspapers and magazines can publish information as they see fit, biased or not. If published information is libelous, the publication can be sued by the person claiming to be libeled. But the federal government does not and cannot regulate newspaper content."