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.