Bugs:

• = is the assignment operator, not a comparison operator. Use == to compare numbers. Use eq to compare strings.

Improvements that can be made:

• Paragraph mode could be used here. Check out $/.
• There's no need to read in the whole file into memory.
• Using 3-arg open when possible is safer. (open(..., '<...') → open(..., '<', '...'))
• open is rather likely to fail. Check to make sure it succeeds.
• Using lexicals when possible is safer. (FILE → $fh)
• Uppercase variable names are usually reserved for constants.

Resulting code:

my $file_name = ...;
my $kwd       = ...;

open my $fh, '<', $file_name
   or die("Unable to open file \"$file_name\": $!\n");

local $/ = "";
while (<$fh>) {
   chomp;
   my @fields = split /:/, $_;
   if ($fields[1] eq $kwd) {
      print "$_\n\n";
   }
}
[download]

Update: You asked about writting it to another file:

my $fn_in  = ...;
my $fn_out = ...;
my $kwd    = ...;

open my $fh_in, '<', $fn_in
   or die("Unable to open file \"$fn_in\": $!\n");

open my $fh_out, '>', $fn_out
   or die("Unable to create file \"$fn_out\": $!\n");

local $/ = "";
while (<$fh_in>) {
   chomp;
   my @fields = split /:/, $_;
   if ($fields[1] eq $kwd) {
      print $fh_out "$_\n\n";
   }
}
[download]