How are the 3 slogans so inherently contradictory? How does the Party get people to accept then as logical?
Paradox and contradiction are the heart of the novel, & the heart of the Party. The answer rests with the paradoxical nature of the Party itself. How it rules is the exact opposite of how its propaganda says it controls. It goes directly into the concept of doublethink, which stands as the foundation for Party rule. Emmanuel Goldstein (in the excerpt from the fictional book) explains this through the word "blackwhite":
But it means also the ability to believe that black is white, and more, to know that black is white, and to forget that one has ever believed the contrary. This demands a continuous alteration of the past, made possible by the system of thought which really embraces all the rest, and which is known in Newspeak as doublethink. Doublethink is basically the power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them.
This is how the Party members can accept such rubbish as logical: they use doublethink. Of course, for the Party itself, it's much easier. Simply put, part of each slogan applies to the Party members (war, slavery, ignorance), while the other (peace, freedom, strength) applies to the party leaders. Thus, according to the Party, "war is peace" means that they continually "fight" wars in order to keep peace at home. During times of war, nations generally unite. Of course, if the people are focused on a common enemy, they are much less inclined to notice how unhappy they are in their own lives. So they make less trouble for their government. "Freedom is slavery" can be thought of in the same way. The slavery of Party members equals freedom for Party leaders. Finally, "Ignorance is strength" can be read "Your ignorance is our strength", again meaning that the ignorance of the people translates into the strength of the government.
The three slogans are inherently contradictory just because they are. They each say something that cannot be true.
I believe that the Party makes people accept them just by repeating them over and over. They also do it by making sure that people think that the Party is the only source of truth.
When you hear something over and over, you often come to accept it even if it seems crazy -- you might just accept it on faith. This is especially true if the person or organization that keeps saying it to you is one that you believe. For most people in 1984, the Party is the only one who has ever told them what to believe so they will automatically believe what it says.