• Do not declare variables in a wider scope than needed.
• Use lexical filehandles and 3-argument open with or die.
• If you want to output frequencies per file, you have to store them per file. That's why the first key is the filename.
• The line is already in the key, no need to store it again to a value.
• You do not have to initialize the counting hash. Just increment it.

#!/usr/bin/perl
use warnings;
use strict;

my %cnt;

for my $file (glob '*.abc') {
    open my $FILE, '<', "$file" or die "$!";

    for my $line (grep / FREQ /, <$FILE>) {
        $line =~ s/ is begin.*//;
        $line =~ s/.* FREQ/FREQ/;
        chomp($line);
        $cnt{$file}{$line}++;
    }
}

for my $file ( keys %cnt) {
    for my $line ( keys %{ $cnt{$file} } ) {
        print "$file $line $cnt{$file}{$line}\n";
    }
}
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