I keep hearing people say that the US is not in a state of declared war. I don't know why they say this.

Congress did declare war. It just didn't use the word "war".
The Sept. 14 resolution said:

"The President is authorized to use all necessary and appropriate force against those nations, organizations, or persons he determines planned, authorized, committed, or aided the terrorist attacks that occurred on September 11, 2001, or harbored such organizations or persons, in order to prevent any future acts of international terrorism against the United States by such nations, organizations or persons."

http://www.thenation.com/capitalgames/index.mhtml?bid=3&pid=19

This is quoted from The Nation, which of course views Bush as a
right-wing imperialist warmonger. But even The Nation concedes that
most of the bombing was authorized by a lawful declaration of war.

Sen. Joe Biden, who says he wrote the war resolution, confirms here that
it was a declaration of war:

M: (Inaudible) Talbot(?). Senator, thank you for this broad gauged approach to the problems we face. My question is this, do you foresee the need or the expectation of a Congressional declaration of war, which the Constitution calls for, and if so, against whom? (Scattered Laughter)
JB: The answer is yes, and we did it. I happen to be a professor of Constitutional law. I'm the guy that drafted the Use of Force proposal that we passed. It was in conflict between the President and the House. I was the guy who finally drafted what we did pass. Under the Constitution, there is simply no distinction ... Louis Fisher(?) and others can tell you, there is no distinction between a formal declaration of war, and an authorization of use of force. There is none for Constitutional purposes. None whatsoever. And we defined in that Use of Force Act that we passed, what ... against whom we were moving, and what authority was granted to the President.

http://biden.senate.gov/~biden/press/release/01/10/2001A24C02.html