However, I am unable to reproduce the same result below.

There seems to be something badly wrong with the code given. But no wonder - even the plain text of this script if full of typos!

For that define a function Sym[expr_,n_] which will symmetrize expr by all permunations of x...

Is this:

1/5 f[x[1]] + 1/5 f[x[2]] + 1/5 f[x[3]] + 1/5 f[x[4]] + 1/5 f[x[5]]

then really the wanted result? Or should the code rather reads like so:

sym[expr_, n_] := Block[{prm},
  prm = Permutations[Table[x[i], {i, 1, n}]];
  Total[expr @@@ prm]/Length[prm] // Expand
  ]

Then:

In[]:= sym[f, 3]

Out[]= 
1/6 f[x[1], x[2], x[3]] + 1/6 f[x[1], x[3], x[2]] + 
 1/6 f[x[2], x[1], x[3]] + 1/6 f[x[2], x[3], x[1]] + 
 1/6 f[x[3], x[1], x[2]] + 1/6 f[x[3], x[2], x[1]]

Well - no idea! Maybe you should ask a more specific question.