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In a recent interview, that Mark Levin had with Victor Davis Hanson, Hanson uses the word psycho-grammatic in reference to how Donald Trump ran his campaign.

My Oxford dictionary does not recognise the word (nor does this website - assuming my spelling is not the culprit).

Is it a legitimate word and what does it mean exactly?

He is using the word "psychodramatic".

"psychodramatic" seems to be in Victor Davis Hanson's idiolect. Here is an article written by him titled "Our Psychodramatic Campuses", with the word used in the article itself. Here is another article by him titled "From One Psychodrama to Another." Here is an article by him in National Review titled "A Psychodrama Too Far" where he also uses it in the article."

Here is the point in interview when he says it.

My transcription:

So Trump seemed to be the only one that was unconcerned what the
Establishment - by that I don't want to be vague - what university
professors said, New York Times said, PBS, NPR, Council on Foreign
Relations; he just spoke to the people. And he did it in such a way
that was refreshing, and he sort of, was, psychodramatic, that this
was an existential war and that we were losing it, and we had to fight
back, and not being worried what people said.

The entire interview is two conservatives lamenting the deterioration of American values that have primarily been eroded (in their view) by the recent pushes by the left to advocate equality for everyone at the cost of the classical American philosophy of classical liberalism and individualism. See "existential war and that we were losing it"

More relevantly, the primary meaning of the word psychodramatic is to do with the psychotherapeutic technique of psychodrama. You don't have to know about this, because this is not the way he uses it in my opinion. If we look at this definition:

psychodrama

3. An event, social interaction, or narrative that manifests psychological forces or problems:
American Heritage Dictionary

psychodrama

an often ongoing psychological struggle also : an expression of psychological turmoil
Merriam-Webster Dictionary

I personally think the use of this term is an odd or obscure flourish on his part, but what I take him to mean is that Trump's message to the people that he would appeal to was ostensibly a message of anti-Establishment leadership or populism, and that this, along with the reaction to it, laid bare Americans' discontent from a collective psychological point of view. By this I mean something like the the national psyche (at least among a significant portion), which harboured a disillusionment with falling or stagnant real wages, growing wealth disparity, employment issues, dissatisfaction with immigrants' effects on the country's economy and society.

To quote again Merriam-Webster:

an often ongoing psychological struggle also : an expression of
psychological turmoil

The psychological turmoil I interpret in this context as the American collective psyche, or as the collective psyche among those who thought Trump was the right person to lead their country.

Edit: Since I've learned that he uses this word quite often, I should point out that my above interpretation is my personal opinion based on the context of the interview, which I did watch in its entirety. However seeing as he uses this word so often it may be difficult to pinpoint his exact meaning without getting to know this man a bit better.