Why Euler's Totient function is important?
A few things:
You reduce many term polynomials, to just a few terms mod any value. (see here as applied to an integer, a less general form of polynomial).
It shreds power towers down to size, with repeated use.
It allows us to work in smaller numbers, rather than potentially trillion digit numbers.
It allows us to pigeonhole principle coprime variable sets.
You can generalize it to products of coprime arithmetic progressions. The product of the first 4 numbers in arithmetic progression 10k+9 have a product that is 1 mod 10 for example (193,401 for those wondering).
Can be used to limit long division in finding the reptend length of fractions with coprime denominator in a given base.
Cryptography ( forgot this)
- etc.
Another reason from abstract algebra:
If $n$ is a positive integer then every group of order $n$ is cyclic if and only if $\gcd(n,\varphi(n))=1$.
You can read about the history and proof of this result here.