I'll try to give a "philosophical" answer:
If you take the so-called "Platonic" viewpoint which many mathematicians seem to share more or less, then mathematics is something which exists in some realm outside of us and can be "examined" by our minds. From this point of view the natural numbers are a reality that everybody experiences in the same way and induction is something that's "obviously true" about them. (Or, in other words, you have to "believe" it.)
If, on the other hand, you take a more formalist point of view and adhere to Bertrand Russell's claim that mathematics is essentially a collection of statements of the form $A \Rightarrow B$, then before you start investigating a subject you try to find a couple of axioms that describe your subject as concisely as possible. If you do it this way, then the principle of induction (or something which is equivalent to it) is essentially always an axiom, i.e. something that can't be proved.
One set of axioms that can in principle be used as a basis for almost all of mathematics (and is what you normally use without thinking about it unless you are either an ultrafinitist or working on foundations) is ZFC which includes an "axiom of infinity" from which the existence (!) of the set of natural numbers and then the principle of induction can be derived.
In some areas of mathematical logic you're specifically interested in "weaker" axiom systems to see what you can (and what you can't) prove with them alone. This is were systems like the Dedekind-Peano axioms (also of great historical interest) come into play which also include an axiom (or sometimes a whole schema of axioms) for induction.
Or, to put it more strikingly: The principle of induction says something about infinitely many objects which means that you can neither prove nor "check" it with finite means (and that's all we mere mortals can do). The only way to "prove" it is to fall back to some other proposition about infinitely many objects which again can't be proved.