You don't know that an axiom is true, you just accept it as if it was true.
People will take the axiom, and other axioms, and logical rules, and make deductions. You then look at the deductions. If the deductions result in contradictions then your axioms are not well chosen.
For example, in mathematics there is the axiom "for any two real numbers a and b, exactly one of the statements a < b, a = b, or a > b is true". If we replaced this with "for any two real numbers a and b, exactly two of the statements a < b, a = b, or a > b are true", then we would eventually end up deductions producing contradictions.
Again in mathematics, we know that there are statements that can neither be proven to be correct nor can they be proven to be wrong. If you have such a statement S, you can take as an axiom. We get a slightly different mathematics. We could have taken (not S) as an axiom, and again get a different mathematics. So in this case, any one out of two contradicting statements can be taken as an axiom.