The Maryland Law Review published an article summarizing several sources of the U.S. freedom of the press (Bogen, 1983).
Parliamentary Privilege
Prior to the American revolution, freedom of press and speech were only applied to members of Parliament as a part of their official duties. At this point the two rights were distinct: members of Parliament established the freedom to openly debate matters of policy and criticize the crown (freedom of speech), as well as the freedom to publish and circulate their own documents (such as laws) without the crown's interference.
However, this only protected Parliament from the executive. It didn't protect citizens from any part of the government. In the colonies (as well as under the Articles of Confederation) American legislatures adopted this standard. Eventually, the Bill of Rights extended this to all Americans.
Censorship & Libel
One of the historical concerns behind the freedom of the press is the threat of censorship and libel. Many of these concerns were directly related to printing technology. In English history, the government had established printing monopolies and strict licensure rules, as well as censorship, to control what could be printed. Freedom the press is intended to prevent the American government from doing this.
Additionally, Parliament had previously sued some printers for libel. Parliamentary privilege at one time prohibited anyone from publishing what was said in Parliament. When people started printing those records, Parliament responded with libel suits. Freedom of the press also addresses this concern, by making it clear that the press can publish the affairs of government.
This isn't mentioned in the article, but some of the founding fathers were printing professionals and would have been familiar with some of these issues.
Press and Speech
The authors of the Constitution and First Amendment considered "publishing" and "the press" to be different things. However, the two concepts are related. Madison said:
"The
people
shall
not
be
deprived
or
abridged
of
their
right
to
speak,
to
write,
or
to
publish
their
sentiments;
and
the
freedom
of
the
press,
as
one
of
the
great
bulwarks
of
liberty
shall
be
inviolable.
In this sense, freedom of speech encompasses both the right of citizens to speak and publish their views. Freedom of the press is specifically the protection of the printing industry from undue government influence.
Jefferson recommended using this language:
The
people
shall
not
be
deprived
or
abridged
of
their
right
to
speak
or
to
write
or
otherwise
to
publish
any
thing
but
false
facts ...
At this point, there is no mention of freedom for the press.