How did the slogan "Liberty, Equality, Fraternity" sum up the goals of the Revolution?

I assume you're talking about the French Revolution, which fed the Haitian Revolution as well.

Louis XIV and his wife Marie Abtoinette were extraordinarily wealthy. To name a few privileges they enjoyed: the majority of taxes went to the Versailles Palace that they lived in, Louis XIV had extravagant feasts for every meal, and the queen was obsessed with frivolous and unnecessary hats. These activities, and Louis XIV's failed wars, contributed to a terrible financial crisis that compelled Louis XIV to tax the poor more and more and more. But they couldn't keep this up because the poor were starving and couldn't pay enough. Their resentment for the proceeded aristocrats increased exponentially.

Louis XIV decided to convene the Estates General to brainstorm better ways to solve his debts. The first two estates were the clergy and the aristocrats, with roughly 90-97% of the population belonging to the 3rd estate. These were peasants all the way to artisans. Before discussing the financial problems, though, they had to agree on how to vote. The first two estates wanted to have one vote for each estate, meaning the top two could always outvote the third, but the third estate wanted to vote proportional to size (head count). Then they would have a greater say. When this debate reached a stalemate, the third estate decided to create their own constitution: The Declaration of the Rights of Man and Citizen, that focused on Equality and how every man deserves to be treated fairly since they all must contribute to society. It outlined duties and rights of every man.

Of course, this was a relatively peaceful development that brought together a "brotherhood". This peaceful Republic later dissolved into the Reign of Terror which employed mass executions and censorship, probably going against fraternity and many other founding principles.

I hope this provided some jumping-off points for an essay!