I really recommend you use proper indentation for your blocks. This will make the error quite obvious. Indent the content of each block by four spaces:

#!/usr/bin/perl
use strict;
use warnings;
my $filename = "filet.txt";
my $emailRE='/(\b[\w\.@]+\b)/';
my @emails=();
my $email;
my $mail;
while (<>) #for each line in each file
{
    @emails = (/($emailRE)/og); #get all the emails on this line

    foreach $email (@emails)
    {
        print "$email, $.\n"; 
        open(FILE, ">$filename") or die "Can't write to $filename: $!"
+;
        print FILE $email;
    } # print them out
    #close FILE;
}
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As a second step, I recommend printing out the progress of your program. For example, add a warn statement when you open your output file.

Also, you may want to learn about the arguments to open. Especially take a look at what > means - it means create-and-overwrite.