For a given $n>1$, let integer $f>0$ be such that for all $m$ in $[0,n)$ with $\gcd(m,n)\ne1$ it hold $m^f\bmod n=1$. One such integer $f$ is the Euler totient of $n$, $\operatorname{phi}(n)$ aka $\varphi(n)$, $\Phi(n)$ or $\phi(n)$. Among so many Euler theorems, the one in the question likely is about that property of the Euler totient. The smallest such $f$ is $\lambda(n)$, where $\lambda$ is the Carmichael function.
Assume $e$ and $d$ have been chosen such that $e\cdot d\bmod f=1$. By definition¹ of what the operator$\bmod$ is when there is no opening parenthesis immediately on it's left, that means: exists integer $k$ such that $e\cdot d=k\cdot f+1$ (and $0\le1<f$, which stands).
Now, assuming $\gcd(m,n)=1$,
$$\begin{align}
\left(m^e\right)^d\bmod n&=m^{e\cdot d}&\bmod n\\
&=m^{k\cdot f+1}&\bmod n\\
&=m^{k\cdot f}\cdot m^1&\bmod n\\
&=m^{f\cdot k}\cdot m&\bmod n\\
&=\left(m^f\right)^k\cdot m&\bmod n\\
&=1^k\cdot m&\bmod n\\
&=1\cdot m&\bmod n\\
&=m&\bmod n\\
\end{align}
$$
We have proven this under the condition $\gcd(m,n)=1$, which is what the original RSA paper does, and many introductions to RSA do. But that happens to be true under a condition not involving $m$: that $n$ is square-free, see this.
This "square-free $n$" condition is much more satisfying than $\gcd(m,n)=1$ in the context of encryption of arbitrary message $m$, especially when we use artificially small $n$, since then we can't summarily rule out $\gcd(m,n)\ne1$ as unlikely. In the question $n=33$, thus $\gcd(m,n)\ne1$ occurs for $m$ one of $0$, $3$, $6$, $9$, $11$, $12$, $15$, $18$, $21$, $22$, $24$, $27$, $30$, thus including the $m=15$ considered!
¹ For integer $s$ and integer $t>0$, equivalent definitions of what the operator$\bmod$ is when there is no opening parenthesis immediately on it's left include
- $s\bmod t$ is the uniquely defined integer $r$ with $0\le r<t$ and $s-r$ a multiple of $t$
- $s\bmod t$ is the uniquely defined integer $r$ with $0\le r<t$ such that exists integer $k$ with $s=k\cdot t+r$
- depending on sign of $s$, $s\bmod t$ is
- if $s\ge0$, the remainder of the Euclidean division of $s$ by $t$
- if $s<0$, either
- $t-((-s)\bmod t)$ if that's not $t$
- $0$, otherwise
This is not to be confused with the notation² $r\equiv s\pmod t$ with opening parenthesis immediately on the left of$\bmod$, which equivalent definitions include:
- $s-r$ is a multiple of $t$
- exists integer $k$ with $s=k\cdot t+r$
² $r\equiv s\pmod t$ is preferably read with any of the possibly several $\equiv$ on the left of$\pmod t$ read as congruent or equivalent rather than equal, and with a pause where the opening parenthesis is. That pause is to mark that$\pmod t$ qualifies what's been said. It's common to use $=$ instead of $\equiv$, to omit$\pmod t$, or omit the opening parenthesis before$\bmod$. That's also a common cause of confusion when the difference between$\bmod t$ and$\pmod t$ matters, which includes computation of ciphertext in RSA.
