Let $\{x_n\}$ be a Cauchy sequence in $M$. Lets see that $\{x_n\}$ is convergent on $M$. This shows $M$ is a complete metric space. By Hopf-Rinow's theorem, $M$ is complete, in geodesic sense.
Claim: $d_{\bar{M}}(y_m,y_n)\leqslant \frac{1}{c}d_M(x_m,x_m)$
Proof: given a curve $\gamma$ in $M$, we have $\ell(\gamma)=\int|\gamma'|dt\geqslant c\int|df_{\gamma}\gamma'|=c\cdot \ell(f(\gamma))$
Once $f$ is a diffeomorfism, the differentiable curves of $M$ and $\bar{M}$ are in bijection. So, considering the curves joinning $x_m$ to $x_n$ (and its images joinning $y_m$ to $y_n$), we can take infimum and it leads us to the result.
Claim: $y_n=f(x_n)$ is Cauchy in $\bar{M}$.
Proof: it follows directly from the claim above.
Hence, once $\bar{M}$ is complete, we have $y_n\rightarrow y$ in $\bar{M}$. But $f$ is a diffeomorphism. Then $x_n=f^{-1}(y_n)\rightarrow f^{-1}(y)$ in $M$.