Forms come from direct analogy to mathematics. The interactions of particles can participate in the form of a triangle. The mathematical person's mind can see that timeless thing, in the transient phenomena.
Plato put music as well as math at the heart of his education system, and that's the other cue. He saw relationships of harmony and resonance as leading to a life that could be beautiful in itself, like a great composition.
Does the issue not arise then, that persons do not autonomously
participate in Forms? That, for instance, you are either courageous or
not, and this is a matter of ontological being.
When you shape your fingers like so, or cross your leg in such a way, do you autonomously participate in triangles? Sure. The fingers or legs don't blink in and out of existence though, they participate in other forms in other ways, before and after the shift. But arranging your fingers in a ratio, or you legs, could impact strength in a certain direction. There are real properties, affected by participation in geometrical relationships.
For Plato, triangles are 'more real' than human bodies. We gain our reality from qualities of geometry. Similarly with transcendental ideas like courage. What ever ontological existence we have exists through participation in the forms.
This can seem very abstract, but I would point to how this idea is still very much alive in the thinking of Mathematical Platonists, like Tegmark with his book 'The Mathematical Universe'. It's common to describe our universe as a space of potential resulting from the forces and particles that exist. Lisi described the set we have as the participation in a fracture plane, resulting from cooling from the E8 hyperobject, the space of all possible particles and forces (in M-theory or String Theory). Sean Carroll describes the 'Core Theory' equation, or there are other ways of framing that, or other candidate equations. This is usually pictured as an ontologically more fundamental or universal layer, which observed phenomena are progressively emergent from, but necessarily constrained by.
Compare this to Plato's model. He identified the Nous (usually translated as "Intellect", or "Intelligence" in this context, or sometimes "mind" or "reason") which is described as God, or more precisely an image of God, often referred to as the demiurge.
"Forms can account only for what it is for each thing to have the
features it has: they can no more account for why a thing has the
features it has at a certain time, or why it is related to other
things that have other features, so that the totality is harmonious
and intelligible, than can the mere postulation of a material cause.
However, nous, understood as an active causal principle which aims at
what is best, does account for these things." -from Stephen Philip
Menn, Plato on God as nous
It thinks its own contents, which are thoughts, equated to the Platonic ideas or forms (eidos). The thinking of this Intellect is the highest activity of life. The actualization of this thinking is the being of the forms. This Intellect is the first principle or foundation of existence. The One is prior to it, but not in the sense that a normal cause is prior to an effect, but instead Intellect is called an emanation of the One. The One is the possibility of this foundation of existence.
"I want to know how God created this world. I am not interested in
this or that phenomenon, in the spectrum of this or that element. I
want to know his thoughts; the rest [is] just details." -Einstein, in
a discussion with Esther Salaman, 1925
This leads to the question of why in a non-theistic non-deist picture, we can expect the universe to be intelligible at all as Einstein took for granted, why a meaningful grasp of it can be expected to fit in our little heads.
But that is a separate topic. I feel that a kind of dualist theism that tacitly makes the 'mind of god' the arbiter of 'objective' reality was deeply woven into our thinking by Plato, and we have not truly purged its ghost, in which task we must focus close attention on why this model is seductively intuitive.