Question

What does President Lincoln mean when he says: “these dead shall not have died in vain?”

Answer 1

What this quote is saying is by continuing the fight the audience would insure the deaths of those who died in battle would not be in vain. If they gave up on the cause of the war, then those who died would have died for a lost cause making their deaths in vain. Because of this, the living must "be dedicated to the great task remaining".

Note: Keep in mind this speech was given at a dedication of the Soldiers' National Cemetery.

Explanation:

When analyzing a quote it's important to keep it in the context of the text and background of the quote. Here is context of the quote:
"But, in a larger sense, we can not dedicate -- we can not consecrate -- we can not hallow -- this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us -- that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion -- that we here highly resolve that these dead shall not have died in vain -- that this nation, under God, shall have a new birth of freedom -- and that government of the people, by the people, for the people, shall not perish from the earth."
- Lincoln