What does the Catholic Church mean when it calls humans co-creators with God? The Catholic Church says that human are co-creators with God.


  To human beings God even gives the power of freely sharing in his
  providence by entrusting them with the responsibility of “subduing”
  the earth and having dominion over it. God thus enables men to be
  intelligent and free causes in order to complete the work of creation,
  to perfect its harmony for their own good and that of their neighbors.
  Though often unconscious collaborators with God’s will, they can also
  enter deliberately into the divine plan by their actions, their
  prayers, and their sufferings. They then fully become “God’s fellow
  workers” and co-workers for his kingdom. (Catechism of the Catholic
  Church, paragraph
  307)


In what manner, capacity, and extent are we co-creators with God, according to the Catholic Church?
God created the nature and men, either directly or through evolution (that's a different question). But men created lots of other things: fire, flint tools, wheel, civilisation... This is one aspect: in some sense God created even technology, but we, men, are its direct creators.

There are other aspects. If you testify about Jesus or just live with God and bear fruits of Holy Spirit and your testimony (by words and/or life) becomes the impulse for someone to convert and believe in Christ, this is mainly work of God's grace and Holy Spirit, but you acted as God's fellow worker in this case.

Of course, not everything we do is a participation on God's work on his kingdom. God gave us free will and responsibility to either gather with him or scatter without him.