"What do we mean from saying death to America?" This is my brief answer as an Iranian.
"Death to America" slogan is (or at least turned into) a kind of sign of dedication (to Iran) among the Iranians, it doesn't mean to conquer Washington as some folks once interpreted, it means that "America! we disagree with your wars, crimes, intrusions, and sanctions on others and ourselves"
However, this is very important to cite that if we say death to America, we exclusively mean the US government, not the US nation as well as the leader of Iran, Mr. Ali Khamenei said once.
"Death to America" means:
We never forget that you have overthrown our PM before the 1979 revolution.
We never forget that you assassinate our scientists.
We never forget that you supported Saddam for attacking us, and caused more than 250 thousand of our people/soldiers to die.
We never forget that you assassinated the father of this nation, General Qassem Soleimani.
We never forget that some of our patients that have the rare diseases are dying (also here) just because we can't import their medicines because of your draconian sanctions. (However, recently we are producing those medicines on our own (If the JCPOA doesn't exist, we can produce more of these medicines))
We never forget that you're lying every second (no exaggeration) about us, about our nation, peaceful nuclear program, beliefs, our government, etc.
We don't forget that 2000 of our soldiers died in our war with ISIS in Syria and Iraq, because of your supporting terrorism there. (also here and here)
We never forget that you have established more than 260 TV/radio/website channels for "free" and in "Persian" to keep up with your psychological war and misinformation against our nation. (While the Americans must pay for even TV channels in the US)
(the reference is in Persian, so I translate it: The director-general of Islamic Republic of Iran Broadcasting(IRIB) said in a report that "more than 260 Farsi satellite TV channels are propagating against the Islamic Republic"
We disagree with your wars and occupation in Yemen, Syria, Iraq, Afghanistan, Palestine...
and so much more things that if I wanna cite, it takes a whole day to write, I just mentioned the most important ones.
This doesn't mean to attack America or something, it means that we fight with this regime and their cruelty till their brutality ends, and in fact "death to America" means death to capitalism, death to Zionism, death to globalization, and political structure of the US that is built on war and sanction, right of the bat.
It pays to add that the slogan "Death to America" is always accompanied by "Death to Israel", "Death to Zionism" and "Death to the UK".
Every year on the anniversary of our revolution, 22 Bahman (February 11), we hold a march (for showing that we're still dedicated to the values and goals of our revolution) and these slogans would be chanted by the people, however, there's variety of situations which we chant these slogans.
"Where does it come from?"
We have a belief from our religion (Shiism) that says:
"Those who are silent when others are oppressed are guilty of oppression themselves"
-Imam Hussain (AS) the son of Imam Ali (AS)
[This sentence is truly the explanation of humanity and actually wants to say that "humans are responsible for what/who they support"]
So, we say death to America to voice our disagreement with the crimes of the US and the Zionists and others in the world, not just because the US is our enemy, that's why I say "We say death to America because of Yemenis, Palestinians, Syrians, Iraqis and other oppressed nations".
Note: Due to the fair criticism of some folks in comments I added more references for each part. (Some of the references are from the American/British news agencies that I don't deny or approve the accuracy of all of their words about Iran)