Dionysius wants to work out just how we relate to God. It turns out that, following Plotinus and Proclus, this happens through ascent. Dionysius is the originator of the purgation-illumination-union triad, and it comes out in both his works on hierarchies.
The hierarchy is the sacred (hier) source or principle (archos) which takes us back to God. The celestial hierarchies are the angels. Dionysius does not think that we pray to angels, or anything like that. The point is that the three groups of three types of angels that he talks about give an idea of the progressive holiness and otherness of God. At the same time, the purgation illumination union triad speaks of the communal responsibility that we have in making the ascent possible.
This needs to be amended somewhat, though. It is not just an ascent. Each choir of angels is a community that is drawing ever closer to God.
The ecclesiastical hierarchy is the same principle, but on earth. Dionysius gives us an idea of how church structure is mystical. This is all based on the nature of liturgy. Liturgy is the experienced, not the learned. The priest goes to God with the people, who have been purified to meet God. The liturgy is one of negation, in the sense that it does not simply relate facts about God (cataphatic theology), but draws the person to God. Dionysius calls on the rites of baptism, the eucharist, and the "oil" (something like eucharist).
Take eucharist, for example. Dionysius is concerned to show the movement of the rite. The hierarch proceeds out from the sanctuary, goes to the farthest point of the nave, and returns (emanation, return). God moves out in creation, and creation moves back in, in response. Eucharist is communion, unity, becoming one. The bishop moves out into the congregation, out of his own contemplation, imparts it to the congregation, and returns without any loss.
Dionysius also makes the distinction between kerygma (that which is spoken forth), and dogma (that which is hidden). The mystical path moves toward dogma, which is the revelation of hidden truth.