If I had time I'd write up a full description and some comparisons of the techniques and idioms. I don't. :( The good news is the Moose docs are pretty darn good. Start on the main Moose page and then walk through as many of the Moose::Cookbook entries as you can.

The + in the +shift->$method statements is probably too idiomatic to ask you to discover on your own.

my $thing = shift;
# ...is equivalent to...
my $thing = shift(@_);
# ...which is, except for altering @_, functionally equivalent to...
my ( $thing ) = @_;
# ...or...
my $thing = $_[0]; # Perl really is your friend though!

# shift without parens can become ambiguous to the interpreter,
# not just the next developer! in some cases like...
my $lookup = $hash{shift}; # not going to shift(@_)
# the unary plus forces shift to be interpreted as the function-
my $lookup = $hash{+shift};

# Quick proof-
perl -le '%a = (1 => 2, shift => 4); print $a{shift}' 1
4
perl -le '%a = (1 => 2, shift => 4); print $a{+shift}' 1
2

# I think in the cases in the prototype code I showed you
# The "+" is not necessary. I use it habitually to visually
# disambiguate shift's usage as a function.

# This all means that
sub o_hai {
    my $self = shift(@_);
    return $self->get_some_val();
}

# ...is the same as...
sub o_hai {
    +shift->get_some_val;
}
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Just so you know, many hackers here will tell you that this is suboptimal terrible style. They'd have a case. I find idiomatic Perl easier to read because it's more terse but it is also more confusing to those who don't know Perl idioms. So...

Moose gets conceptually hairy; ask someone to explain the finer points of traits versus roles and the dangers of multiple inheritance to get a feel for that. Moose itself is a joy and, I argue, the clearest path through this problem space and I highly recommend it.