If trying to compare the slogan "liberty, equality, and fraternity" to the goals of the revolution, it is best to look at each of those terms individually.
- Liberty--kind of saying the freedom from oppression and the ability to make one's own choices in life. This ties in with the revolution because that is a major reason it was fought...to have the ability to self-govern without being told what to do by a king and parliament that was across the ocean. It was also freedom from things the colonists found to be offensive, such as rules regarding the ability of the government to obtain "blanket warrants" to search houses on little more than suspicion.
- Equality--this is a tricky one because it is a broad word. Equality sounds like it would mean "for everyone" but that is not necessarily true. Women and slaves, for example, were not held to be equal. Men without property were not necessarily equal. In the sense of the revolution, it is probably taken to mean that men are "equal" to each other and that no one should have a royal title and think they are "better" than everyone else or have special privileges. That was a goal of the revolution, to take the monarchy out of the equation and institute a semi-democracy.
- Fraternity--this usually means "brotherhood" or camaraderie. This occurs between soldiers fighting in a war, but in the sense of the revolution as a whole it more likely refers to a fraternity between the colonies...that the separate colonies were now going to operate much more closely and with a common interest in mind instead of acting like a lot of different little states. That was another goal of the revolution, to tie together the separate colonies and form one stronger entity.
Liberty included freedom from the King and the estate.
Equality mean no more aristocracy and Estate Classes. Everyone should be equal with no other standing.
Fraternity was the slogan to unity between all of the French people