How does "each" change "are" to "is"?

Relevant: 'Each' with plural or singular verb and What should I use between “triple” vs. “all”?

The answers in the linked question don't quite help me. Specifically, what happens with this case:

My three pens are green.

Each of my three pens are green.

Each of my three pens is green.

Part of the confusion seems to stem from the inclusion of "three":

My pens are green.

Each of my pens are green.

Each pen is green.

There doesn't seem to be a way to rewrite the last example with "three":

Each three pen is green.

The more common application would be:

All three pens are green.

But in the event that I want to say, "Each of my three pens are green" which is correct?

And, in the event that the correct sentence is "each of my three pens is green" why does prepending "each of" to the beginning of "my three pens are green" change the verb?

Each of my three pens is green is correct.

You're thinking of each of as something that gets tacked onto the beginning of the noun. But when you add each of to my three pens, my three pens becomes the object of the preposition of. Each becomes the head of the subject. So the verb agrees with that, and each is always singular.

Of pretty much always does this:

The rumors were that she would leave him.

The gist of the rumors was that she would leave him.

Some nouns can be plural if they are followed by an of phrase with a plural object:

A number of flaws / A bunch of flaws / A total of ten flaws was found. (iffy)

A number of flaws / A bunch of flaws / A total of ten flaws were found. (ok)

but each isn’t one of them.

The meaning of each X is something like, “This sentence is true for every individual X”. This is clearest when you look at a sentence that is true of each individual, but not necessarily true of the group as a whole:

The roses cost $24. Each rose costs $2.

"Each of my three pens" means "every single one of my three pens". The phrase "one of my three pens" would take the verb "is" after it, because "one" is singular. Similarly, "each" is singular.

You can remove "of my three pens" to make it even more clear:

Each is green.

If you know the context, then that's the same as saying:

Each of my three pens is green.

This is also the same as saying:

My first pen is green, my second pen is green, and my third pen is green.

Each is intended to enumerate the objects (the pens) so that you mentally think of them individually. You wouldn' t say "My first pen are green," so therefore you don't say "Each pen are green."