I was flipping through the Summa today (in hard copy so I don't have a link), and as I can reconstruct it this is how Aquinas' argument proceeds.
First, since God is one Divine Essence, differences within Him can only be differences of internal relation.
Secondly, there are three possible principles in a living being: The Sensitive Principle (Body), the Appetitive Principle (Will), and the Rational Principle (Intellect).
The sensible principle relates to things when external objects act upon it (e.g, I get hot if you put my skin against something hot). Nothing external can act upon God, so He has no sensitive principle.
The appetitive principles relates to things in so far as the will draws it towards what it naturally desires.
The rational principle relates to things by producing within itself a similitude of them, and this similitude is knowledge.
Sensible, appetitive, and rational are opposite ways of relation (acted upon, drawn towards, producing a similitude of), so they exist within the essence of one living essence. Again, within God there are only the appetitive and rational principles.
Since the only differences in God are differences of relation, the persons are distinguished by opposite relation. For example, the Father relates to the Son and He relates to the Spirit, but since He doesn't relate to them in opposite ways, He remains one person of the trinity.
The realm of the rational principle is the production of similitude (in the case of God's perfect knowledge of Himself, a perfect similitude), and two opposite relations inhere within: Paternity, the relation by which something produces a similitude, and Filiation, where the similitude is considered in contrast to the thing which has produced it. Herein lie the opposite relations that distinguish Father from Son.
The realm of the appetitive principle is that of desire. There are also opposite relations here, of the same type: Spiration and Procession. If Bill loves margaritas, we could say that the desire for margaritas proceeds from Bill, or that Bill spirates the desire for margaritas. However, things can be desired by the will only insofar as they are known by the intellect: If Bill has no conception of what a margarita is, then he cannot spirate a desire for margaritas. Thus, the Holy Spirit cannot be distinguished by opposite relation from either the Father simply or from the Son simply, but only from the Father and the Son, taken together.
Thus, Aquinas concludes that it is proper to say that the Spirit proceeds from the Father and the Son, since the Spirit can only be distinguished from the Father and Son taken together, or from the Father through the Son, since it is through knowledge that the knower desires what is known.