t may be more clear if we reformat your awk code:

{

  s = "";

  for (i = 1; i <= NF; i += 2)

  {

    s = s ? s FS $i : $i

  }

  print s

}

First we have a loop over all odd numbers from 1 to NF inclusive. Inside that, we build a string s. Each loop iteration, if s is not empty we append FS $i; if it was empty we simply assign $i. The net effect of all this is to make s contain the text from all the odd-number fields, separated by FS. Finally, we print s.

A more concise form, without the ternary, would be:

{

  s = $1;

  for (i = 3; i <= NF; i += 2)

  {

    s = s FS $i

  }

  print s

}