The following modification to the regular expression gives your desired output (I took no assumptions about the format of the text following a timestamp):

#!/usr/bin/env perl 
use strict;
use warnings;
use Data::Dumper;

my $initialStr = "[20150302 22:01:05] [1, 2, 3, 4] String0\n [20150302
+ 22:01:05] [1, 2, 3, 4] String1\n [20150302 22:01:05] [1, 2, 3, 4] St
+ring2\n [20150302 22:01:05] [1, 2, 3, 4] String3\n [20150302 22:01:05
+] [1, 2, 3, 4] String4\n [20150302 22:01:05] [1, 2, 3, 4] String5\n [
+20150302 22:01:05] [1, 2, 3, 4] String6\n [20150302 22:01:05] [1, 2, 
+3, 4] String7\n";

my @matches = $initialStr =~ /\[\d+ \d{2}:\d{2}:\d{2}\]\s+(.+)\n/g;

print Dumper \@matches;
[download]

This assigns the text after the timestamp and all following whitespace to an element of @matches.

As to why the following regular expression doesn't work:

my @matches = ( $initialStr =~ /\d+:\d+:\d+/ );
[download]

What happens here is $initialStr =~ /\d+:\d+:\d+/ is being evaluated in scalar context because of the parentheses making it a single element of a list. Therefore, the result you get is the number of occurences of the given pattern in $initialStr and since /g is not used it's either 0 or 1.

Furthermore, to capture matches into an array, which is what you're trying to do here, you need to add capture groups into the regular expression.