This argument is the opposite of William Paley's watchmaker argument. To understand it, you must first read the article on Aristotelian teleology.
II. Non-Intelligent Things Act for an End
We look around us in the world and see that many things aim at specific effects, rather than just any effect, as outlined in the article on Aristotelian teleology. These effects are often aimed at by things that are non-intelligent: volcanoes aim at venting magma, rivers aim at draining highlands, plants aim at taking in nutrients and absorbing the sun. In other words, there is regularity in the universe.
III. Accounting for Final Causality
Since non-intelligent things around us act for a specific end, aim, or goal, then there must be an explanation for this. For in a mechanistic world (see the article on Aristotelian teleology), any effect could follow in theory from any cause, since mindless matter is not aiming at any specific effect. The only way that matter could aim at specific effects is if directed to do so by some intelligence. Just like the construction crew aims at the specific end of building a building, but only because they are directed to do so by the architect.
IV. Existence Itself
Whatever it is that directs things to their end is also the thing that determines their form and structure. I.e., whatever puts their essence together with existence, as in the Second Way. And so the agent that determines the final cause is also the agent whose essence is existence itself, and thus we are brought to pure existence, or pure actuality.