I have considered for a long time how to best and most simply explain the critical elements of intersectionality, because I think it is among the most critical of ideas and so many people get it wrong. So here we go.
Intersectionality, at it's core, asserts that identities can not be broken into what we see as their individual parts, in particular when discussing the oppression and prejudice a person faces. Identities are more than just the sum of their parts.
The theory was originally laid out by a black woman to explain her own oppression. In short, she claimed that her oppression as a black woman could not be simply explained as black oppression + woman oppression. She also experienced black woman oppression.
Now, the nuances of the situation are much more complex, but this is the very basic idea. You always have to account for a person's identity as a whole, and not just the individual parts.
Black women are not just "black" + "women"
Black men are not just "black" + "men"
Trans women are not just "trans" + "women"
Trans men are not just "trans" + "men"
You always have to consider the identity as a whole, which may mean there are non obvious aspects of the whole identity that are not readily explained by the sum of the individual parts.
That is the basic explanation. Now let's get into one particular application I think is very important, at least on Tumblr, because on tumblr it has become extremely controversial.
A consequence of intersectionality is that specific kinds of oppression cannot be divorced from maleness. A black man's oppression cannot be explained as "black" + "male" oppression. There are aspects of the oppression black men face that cannot be separated from their identity as a whole. Even if you believe "men" suffer no unique oppression, intersectionality means you can't deny the possibility that "black men" suffer oppression that cannot be divorced from them being men.
This application of intersectionality is fundamental and the application of intersectionality to black men has been accepted for basically as long as intersectionality has been around.
A similar application can be made to trans men.
Now if you don't agree with this model, sure. Im not going to say it is fine because I think you are wrong, but no social model is above criticism. What you can't do is claim that intersectionality is anything else. You either accept that aspects of oppression can be tied to maleness, or you do not accept intersectionality. There is no secret third thing. This is a fundamental consequence of intersectionality as a model.
I think it is important to note that intersectionality did not start as queer theory. The roots of the model are in the study of sexism and racism. This is black feminist theory. And to put a point on it, trans feminist theory has no business trying to warp intersectionality to fit preconceived ideas about how queer identities and queer oppression works.
I believe that twisting well established black feminist theory into an unrecognizable form and then stealing the credibility of that well established black feminist theory by labeling your new theory "intersectional" is racist. We do not get to borrow their credibility while spitting on their work.
Again, if you do not believe intersectionality is correct, or that it does not apply to queer identities, that is an argument you can make. I would recommend you have a lot of very well sourced data backing up a radical claim like that, but no model is above criticism. But don't try to trade on the credibility of black feminists while disregarding their work.