I'm trying to undestand when an elliptic operator on a manifold is bounded.
Any elliptic operator in a compact space is Fredholm (see for example this question) In this wikipedia article it specifically states that any Fredholm operator is bounded by definition, so if this is correct any elliptic operator on compact spaces should be bounded as well.
This looks kinda strange for me, so i am wondering if this result is true and if there is a simpler way to prove it (or a counterexample)
First of all you need to be precise with the domain and codomain of the operator. The statement you cited is referring to the Sobolev spaces $W^{r,2}(M)$, $2$ being the Lebesgue integrability parameter.
To see what it's happening, set $M = \mathbb{S}^1$ and $$A f = \frac {d^2}{d t^2} f$$ which is clearly elliptic.
Using the Fourier transform we see that $A$ is conjugated (via an isometry) to $$\sum \hat f_k e^{ik t}\mapsto \sum -\hat f_k k^2 e^{i k t}$$ modulo some constants irrelevant to my argument.
The Sobolev norm of $\vert\vert f\vert \vert _{r,2}$ may be defined using the Fourier transform as $\big(\sum \hat f_k (1+k^2)^r\big)^{1/2}$ now you can check easily that $$\vert \vert Af\vert \vert_{W^{r-2,2}} \leq \vert \vert f\vert \vert_{W^{r, 2}} $$
consequently $A: W^{r, 2}(\mathbb S^1)\to W^{r-2, 2}(\mathbb S^1) $ is bounded.