I think it is pretty straight forward:

if (-e $name) {

basically check if the file exists or not. And you know about this already I suppose.

-e ($name = $name."_$i")

in the for loop basically loop over the file. It has the same meaning as if you do this:

    my  $name =  $name."_$i";        # input file name

    open ( INF, '<', $name )
        or croak "$0 : failed to open output file $name : $!\n";

    while(<INF>) {

     # Do something
     }
    

    close ( INF );         # close input file
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Finally, read -X.