The short answer: delegates of slave states at the Continental Congress approved the Declaration of Independence in its current form, because it was already significantly edited from Jefferson's first draft. This was as good as it was going to get, and they wanted to stop arguing about the Declaration's draft and start fighting the Revolutionary War.

Pop representations of the Declaration of Independence's authorship and signing tend to downplay the amount of committee work and negotiation that went into the final draft. Danielle Allen's Our Declaration is a smart, accessible overview of the process behind the text and exactly how many pieces were sacrificed as the parties got to yes.

First, the Continental Congress had to decide to write a declaration of independence-- e.g. achieve a majority vote in favor of Virginian Henry Lee's call for independence. Not every delegation received a unanimous internal vote, and the New York delegates actually had to return home to re-meet with New York Convention and receive directions about voting. See, contemporary US pop history tends to gloss over the fact that not everyone in the colonies wanted independence. Most historians accept a conservative 20% estimate of colonists who publicly declared themselves Loyalist and provided aid and money to British forces. The percentage of colonists we'd call today "undecided" was higher than that-- somewhere between 20% and 30%. Robert Calhoon estimates that only 40-45% of the population supported independence and the Revolutionary War. So the delegates went through a lengthy internal negotiation process, trying to determine how to represent their states' interests, but also trying to persuade and come to agreements with other states' delegates. It took four days of discussion before the Congress came to an agreement.

Once the Continental Congress had a majority vote to commission a Declaration of Independence, they then had to establish the Committee of Five who would oversee the drafting. This emerged in tandem with the discussion about who the primary author should be. The delegates initially pushed for Richard Henry Lee, who'd penned the initial resolution. If he wasn't also busy writing the Articles of Confederation, and if his wife hadn't suddenly fallen ill, we might have had a very different Declaration of Independence.

As it was, Jefferson himself wanted to serve on the Committee of Five and represent Virginia's interests, but didn't want to write the Declaration himself-- he pushed for John Adams. Benjamin Franklin and Adams persuaded Jefferson to take the lead. Adams recounts in a letter to Timothy Pickering:

Jefferson proposed to me to make the draft. I said, 'I will not,' 'You should do it.' 'Oh! no.' 'Why will you not? You ought to do it.' 'I will not.' 'Why?' 'Reasons enough.' 'What can be your reasons?' 'Reason first, you are a Virginian, and a Virginian ought to appear at the head of this business. Reason second, I am obnoxious, suspected, and unpopular. You are very much otherwise. Reason third, you can write ten times better than I can.' 'Well,' said Jefferson, 'if you are decided, I will do as well as I can.' 'Very well. When you have drawn it up, we will have a meeting.'

Then Jefferson had to finalize his own draft and submit it to the Committee. Adams and Franklin suggested some minor but crucial changes (for example, correcting "we hold these truths to be sacred & undeniable" to "we hold these truths to be self-evident", and speaking of American "citizens" rather than "subjects"). Jefferson made these changes and re-submitted to the Committee of Five, who had thirty-two similar edits involving word choice, structure, and the addition or deletion of sentences. This draft was approved and submitted to the Continental Congress for review...

And promptly ignited a firestorm of controversy. The major issue was Jefferson's construction of the British monarchy as a tyranny opposed to natural freedom, and American slavery as a natural result of that tyranny. In the now-famous major deleted portion of the Declaration, Jefferson writes:

"[King George] has waged cruel war against human nature itself, violating its most sacred rights of life and liberty in the persons of a distant people who never offended him, captivating & carrying them into slavery in another hemisphere or to incur miserable death in their transportation thither... Determined to keep open a market where Men should be bought & sold, he has prostituted his negative for suppressing every legislative attempt to prohibit or restrain this execrable commerce.

Jefferson anticipates that the crown will incite slaves to rise up against their American masters in exchange for freedom

"purchase that liberty of which he has deprived them, by murdering the people on whom he has obtruded them: thus paying off former crimes committed again the Liberties of one people, with crimes which he urges them to commit against the lives of another."

In this construction, American slaveholders and enslaved Africans are both suffering at the hands of the British! Twas the tyrannical King who brought the slaves to America and refused to let the states legislate against slavery! He stole liberty from both white Americans (who frequently used slavery metaphors to describe their political situation) and African slaves. Moreover, slavery is explicitly condemned as a crime against nature and proof of tyranny in action. In this draft of the Declaration, permitting slavery in the colonies would be a crime against the laws of Nature and Nature's God, and would make the new government no better than tyrannical King George.

Delegates of slaveholding states pushed back against this. The Virginia Convention had already forced a similar revision two years earlier, in the Declaration and Resolves of the First Continental Congress. The Congress resolved that inhabitant of the English colonies of North America (so already excluding indigenous peoples) was entitles to "life, liberty, and property." The Virginia Convention refused to ratify this language "on the ground that it was not compatible with the principles of a slaveholding society." Just as in 1774, the Virginian and Carolinian delegates refused point-blank to approve the Declaration until this passage was removed. John Adams writes that the Continental Congress has devolved into a battle of speeches and impassioned debates about slavery's compatibility with the Enlightenment principle of human equality:

[The process] is halted amidst flights of oratory... especially that concerning Negro slavery, which, though I knew [Jefferson's] Southern brethren would never suffer to pass in Congress, I certainly would never oppose.

Multiple other delegates, including Robert Treat Paine (PA) and William Ellery (RI) write to their conventions, complaining that controversy and debate is impeding the approval of the Declaration. Delegates of slaveholding states, like Thomas Nelson Jr., a Virginian delegate and wealthy planter, steadfastly opposed the inclusion of antislavery language. Jefferson, Thomas Paine, and others in their coterie who were wedded to Enlightenment philosophies, equally steadfastly refused to take out the Lockean link between political equality and naturally-bestowed human rights.

Because the Declaration of Independence had to be unanimously approved-- every delegate of every state-- compromise was inevitable. Slaveholding delegates approved the final document because, by and large, they did believe in equality-- just a more qualified version of it. They wanted to be equal to the English, they wanted their statesmen to be equal to the citizens of other states, and they wanted to dissolve class boundaries and declare equality with noblemen, royal-born men, and wealthy merchants.

Jefferson himself went on to write Notes on the State of Virginia, where he concludes that nature has set firm boundaries between racial categories-- so to follow the laws of Nature means equality within a racial group. Others reconciled the tension between the practice of slaveholding, and their desire for political independence from Britain and equality in relating to the other states, by deciding that "men" did not include "subhuman" laborers. But all the delegates had a sense of urgency. They knew that declaring independence meant fighting a war, and needed to move fast.

The final document, the text that we read today, reflects this legacy of compromise, deliberative procedure, and tension between political ideals and lived realities.

Edit: fixed bolds to italics

I asked a similar question a while back and got some great answers. Check it out here: https://www.reddit.com/r/AskHistorians/comments/4qxzdx/how_did_thomas_jefferson_reconcile_all_men_are/

I didn't see it mentioned in the response by /u/0hmyscience, but part of the answer would have to be in the definition of "men" at the time. This was not universally interpreted as "all human beings," but was instead, essentially property-owning males.

The "Three-Fifths Compromise" at the Constitutional Convention in 1787 literally specified that slaves were explicitly NOT equal in that they were counted as "3/5" of a person for the purpose of determining political representation and apportionment.

In part, while the Declaration of Independence and the Preamble are beautifully written documents, there is a ton of ambiguity in phrases like "all men are created equal." In part, this is to reflect the ambiguity of politics and rights at the time, and in many cases the ambiguity was necessary to secure the near-universal acceptance of the big radical ideas like separation from English authority. "Compromises" helped defer the conflict over basic questions of WHO was equal, HOW equal they were, etc.

I don't want to delve too deeply into the history of slavery and racial inequalities, as that is generally well covered, but the linguistics of what the words in the foundational documents of the United States are pretty fascinating. Both the Second and the Ninth Amendments to the Constitution reference the "people," and even now, there is not universal interpretation of the inclusiveness of that word. (For example, even today a question like "should undocumented aliens with criminal convictions enjoy the right to buy guns?" has contradictory court rulings on it). It wasn't until the Fourteenth Amendment (in 1868... a LONG time after "all men are created equal") that the law of the land was explicitly "all men" and there were a number of court cases clarifying the exact meaning of this in the last couple decades of the 19th century.

Then, moving into the 20th century, you have "all (wo)men" added to the mix in 1920 with the Nineteenth Amendment, various political strains on the concept during World Wars I & II (e.g. "all men are created equal... except the Japanese.") and a variety of back and forth on the topic of equal protection in court decisions regarding civil rights in the 1950s and 1960s, etc.

So, short answer, slaveholders AND abolitionists signed off on "all men are created equal" because the document was targeted at unyoking the United States from English monarchal rule, not establishing a clear definition of what rights "men" would have in the United States and who fit into the definition of "men."