I'm assuming you're referring to the regex in GrandFather's reply:
    /data-src-hq="([^"]+)"/g

First, let me draw your attention to YAPE::Regex::Explain, which can explain regexes that do not have regex operators or features added after Perl verion 5.6:

c:\@Work\Perl\monks>perl
use strict;
use warnings;

use YAPE::Regex::Explain;

print YAPE::Regex::Explain->new(qr/data-src-hq="([^"]+)"/)->explain;

__END__
The regular expression:

(?-imsx:data-src-hq="([^"]+)")

matches as follows:

NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  data-src-hq="            'data-src-hq="'
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    [^"]+                    any character except: '"' (1 or more
                             times (matching the most amount
                             possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  "                        '"'
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

There are also on-line regex explainers.

Now let me address your narration.

img src=" is the first part to match

Ok.

[^"] match everything but a quote

I would word this as match a single character from the class of all characters except a  " (double-quote). It's important to realize that the  [...] regex operator defines a character class or set (see Character Classes and other Special Escapes in perlre and also this topic in perlretut, perlrequick and perlrecharclass), and that all by itself, any  [...] matches only a single character.

+" stop when you hit a quote

I would quarrel with this description. The  + quantifier (see Quantifiers in perlre; see also the topic of quantifiers in perlretut and perlrequick) is associated with the expression before it, i.e.,  [^"]+ and I would read it as match one or more characters from the class/set of all characters except a double-quote. Again, the double-quote is not directly associated with the  + quantifier in your  +" — but see below because they are | can be related.

() return only what matches within the brackets

Ok.

Am also curious what's the difference between +" and +?" since both seem to work

Again, note that the  + or  +? quantifiers affect the preceding  [^"] character class, not the double-quote that follows. In the  /data-src-hq="([^"]+)"/g match regex, the final  " (double-quote) is not absolutely needed because  [^"]+ will match as much as possible until it either hits a  " or the end of the string. (I would still tend to use it because I like the feeling of security that well-defined boundaries give me. Also, a final " in the match will prevent a match with a "runaway" quote in a string in which the closing " is missing.) However, if you use a  [^"]+? "lazy" or "non-greedy" expression instead, the final " becomes vital to matching the entire contents of the double-quoted substring. Try this:

c:\@Work\Perl\monks>perl
use strict;
use warnings;

my $s = 'foo "xyzzy" bar';

print qq{+? (lazy) quantifier with final ":    matched '$1' \n} if $s 
+=~ /"([^"]+?)"/;
print qq{+? (lazy) quantifier without final ": matched '$1' \n} if $s 
+=~ /"([^"]+?)/;

__END__
+? (lazy) quantifier with final ":    matched 'xyzzy'
+? (lazy) quantifier without final ": matched 'x'
[download]

A lazy quantifier matches the minimum necessary for an overall match. A final " in the regex is necessary in this case to capture the entire quoted substring. Take a look at this and be sure you understand what's going on, i.e., the difference between lazy and greedy matching.