I'll go with Mark's explanation. YAPE::Regex::Explain says:

use YAPE::Regex::Explain;
print YAPE::Regex::Explain->new(qr/^([01])+$/)->explain();
[download]

output:

The regular expression:

(?-imsx:^([01])+$)

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):
----------------------------------------------------------------------
  ^                        the beginning of the string
----------------------------------------------------------------------
  (                        group and capture to \1 (1 or more times
                           (matching the most amount possible)):
----------------------------------------------------------------------
    [01]                     any character of: '0', '1'
----------------------------------------------------------------------
  )+                       end of \1 (NOTE: because you're using a
                           quantifier on this capture, only the LAST
                           repetition of the captured pattern will be
                           stored in \1)
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

So the ()+ match you do puts the last match in $1.