To participate is as it were to take a part, and therefore when something receives in a particular way what pertains to another, it is generally said to participate in it, as man is said to participate in animal, because he does not have the notion of animal according to its full extension; for the same reason, Socrates participates in man. So too the subject participates in its accident, and matter in form, because the substantial or accidental form, which of its own notion is common, is determined to this or that subject, and similarly the effect is said to participate in its cause, especially when it is not equal to the power of its cause, for example when we say that air participates in the light of the sun because it does not receive it with the brightness the sun has (St. Thomas, Exposition of Boethius's On the Hebdomads, 2 - p.147f).Participate: "to take or have a part or share in; partake in; share" (source)
When the Catechism, with St. Athanasius, says "By the participation of the Spirit, we become communicants in the divine nature. ... For this reason, those in whom the Spirit dwells are divinized" (1988), it does not mean that we become God. No. we are "grafted onto the vine which is himself" (ibid.) so that we become partakers of the divine nature (2 Pet. 1:4).
I suspect that additional posts on this topic will follow as or when my understanding of it grows.