That's saying that the class' instances have a method add_things which when called will push its arguments on to the end of the arrayref instance variable in question. It's sugar for something like this:

package Foo;
use Moose;

has 'some_method' => (
  is => 'ro',
  isa => 'ArrayRef',
  default => sub { +[] },
);

sub add_things {
  my $self = shift;

  push @{ $self->some_member }, @_;
}
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The provides makes it write methods which perform that operation (be it push, or pop, or keys, or . . .) on the instance's member in question.

Update: Minor wording tweaks and removed metaclass declaration from sample code since it's superfluous.