In order to figure out how one can assume validity of axioms let's stand back from Mathematics for a moment and look at natural sciences.
How does one come up with Laws of Physics? Why does one assume that, say, energy is conserved quantity? The answer is that physicists don't make any such assumptions. Physicists experiment with a variety of mathematical models, compute what predictions these models would make, and decide whether tentatively to accept those models based on the quality of their fit to the empirically observed reality.
For example, physicists observe that the law conservation of energy holds true in all known circumstances, despite earnest effort to refute it. More importantly, the assumption that the law holds in unknown circumstances leads to valuable insight into the nature and new discoveries, and the law's predictions always prove true when the new techniques make direct experimentation with new circumstances possible. Based on that, physicists accept conservation of energy as a fundamental law; however, they remain open to review that law if new discoveries would disagree with it.
Something similar happens with the foundations of Mathematics. Despite the popular belief imposed by the castrated presentation of mathematics in American schools, mathematicians don't apriori assume the validity of axioms. Instead, mathematicians experiment with a variety of sets of axioms and figure out which minimal set would lead to better understanding of mathematical objects.
According to V.I. Arnold, "Mathematics is part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." Axioms are not apriori assumptions; in Kant's language they are analytic aposteori propositions.
To put it more succinctly, axioms to mathematics are what fundamental laws are to physics. And, just as physicists are open to review of their fundamental laws if new evidence would lead to doubt of their validity, mathematicians are open to review the axioms if new consideration would prompt them to do so. Perhaps the best known case the revision of the Euclid's 5th axiom that led to development of spherical and hyperbolic geometries and later to Riemannian geometry that has become the foundation of General Relativity. Another major revision was done to set theory when Russell has shown that the naive set theory axioms led to contradictions.
The strength of axioms is not in that they are assumed true; on the contrary, their strength is in they have been virtually proven to be true by the overwhelming amount of powerful conclusions that can be made on the basis of their assumption as well as continuous earnest attempt to undermine them and willingness to review them if necessary.