Note that this involves some pretty advanced concepts. The theorem is an observation about how coprimes and modular arithmetic behave.
If a and b are coprime to each other, then there are no common (prime) factors between them other than 1. For example, 4 and 9 are coprime as 4 has factors (2,1) and 9 has factors (3,1).
phi(n) is simply the number of integers from 1..n that are coprime with n. This also tells you how many units are in the ring Z/nZ, which can be though of as the set of integers from 1..n. Let’s take the example n=9. There are the following coprimes:
1,2,4,5,7,8
for each of the above, there is another number where x*y=1 mod 9. For example, 2*5=10=1 mod 9 and 8*8=64=1 mod 9. There is always another number (between 1 and 9 ) that will make it loop around to 1 (or more precisely, equal to the identity element). These are called units.
Euler’s Theorem is that a raised to the power of phi(n) works the same way as long as a and n are coprimes.
If you want me to explain why that is, I haven’t a clue. This is very advanced math and I took only one class of abstract algebra in college a long time ago. Give MathWorld a try.
If you don’t know what a group or a ring is, then whoever is teaching you this is not expecting you to understand why it works; they’re just expecting you to use it like the example shows. And since you tagged the question “calculus”, then I suspect the latter is the case.
(btw, if you can edit the topic tags, replace calculus with “abstract algebra” and/or “number theory”)