How can we prove that the scrambled G matrix in McEliece cryptosystem preserves the minimum distance properties of G matrix?
$P$ is required to be a permutation matrix. Thus, the effect $P$ has on $G$ is simply to reorder its columns. Crucially, this does not change the weight of any of the codewords. Note that this holds irregardless of the code; be it Goppa or something else. 

More concretely, suppose $G$ is any $(n,k,d)$ code. This means that $G$ can correct up to $\frac{\lfloor d \rfloor}{2}$ errors. So let $d' \leq \frac{\lfloor d \rfloor}{2} $ be the weight of $e$. Now suppose 
$$c = mSGP + e.$$
On decrypting $c$ we first apply $P^{-1}$ to $c$ to get
$$ c' = cP^{-1} = (mSGP + e)P^{-1} = mSGPP^{-1} + eP^{-1} = mSG + e'. \tag{1}$$
You see that the r.h.v. in (1) is simply a codeword in the code generated by $G$ with an added error of $e'$.
But the inverse of a permutation matrix is also a permutation matrix, so the vector $e'$ also has weight $d' \leq \frac{\lfloor d \rfloor}{2} $.  Consequently, we can apply the decoding procedure of the code to get rid of $e'$ and obtain $mS$. Since $S$ is invertible, we can obtain $m$ by multiplying with $S^{-1}$ on the right.