Well let's see what your program does:

$ perl -le'
use warnings;
use strict;

print "Enter number : ";
chomp( my $num = <STDIN> );
print "\$num = $num";

my @nums = split /(?<=[\d])/, $num;
print "\@nums = @nums";

my $i = 0;
for ( @nums ) {
    my $regex = qr/[$i]{$_}/;
    print "\$regex = $regex";
    print $num =~ $regex ? "true" : "false";
    $i++;
    }
'
Enter number : 
2020
$num = 2020
@nums = 2 0 2 0
$regex = (?-xism:[0]{2})
false
$regex = (?-xism:[1]{0})
true
$regex = (?-xism:[2]{2})
false
$regex = (?-xism:[3]{0})
true
[download]

On the first pass you are trying to match '2020' with /[0]{2}/, or in other words with /00/.    Since '2020' does not contain the string '00' the program prints false.

On the second pass you are trying to match '2020' with /[1]{0}/, or in other words with //.    Since '2020' contains five places that will match the program prints true.

$ perl -le'$x = () = "2020" =~ /[1]{0}/g; print $x'
5
[download]

The next two follow the same pattern as the first two.

Let's look at the  /[$i]{$_}/ regex. The  [$i] defines a character class that matches whatever  $i interpolates as a string into the regex. In this case,  $i never changes: it is always the number 0 which interpolates as the '0' (zero) character, so the character class matches a '0' character and nothing else.

The  {$_} quantifier of the  [0] character class (which is the same as '0') takes on the values that  $_ interpolates into the regex on each iteration of the for-loop. These values are from the  @nums array and happen to be  (2, 0, 2, 0) successively.

So the  [$i]{$_} regex expression alternates between  0{2} and  0{0}. That's all there is to the regex. The  0{2} regex does not match (twice!) against the  $num string of '2020' because that string does not have two consecutive '0' characters anywhere in it. The  0{0} regex does match (again, twice) because there is a place in the '2020' string where there is no '0' character.

See perlre, perlretut, perlrequick.

Update: Rats. Missed the  $i++; statement. Of course, the analysis of jwkrahn is correct, while mine is wandering up and down in the street outside the ballpark. .oO(But was  $i++; present in the original original post, the one to which I replied? The more I think about it, the more I think it wasn't!)