To roughly summarize, Intersectionality is a general way of saying "there is an underlying thought process that forms the scaffolding of all oppression." I.E. everything from Racism, Sexism, what have you is entirely the same process created by a thought that has the same root and thus ALL activists fighting oppression can use intersectionality to unify all these struggles and thus all struggles can create a united front against the Oppressor Class. Essentially there is an interconnection between all oppression. Ignoring any plausible flaws in this way of thinking, it is a very potent way for everyone from Feminists to Vegans to rally into a single cohesive movement for mass solidarity. So what is the problem? Also I freely admit this is a very crude paraphrasing. If I am wrong feel free to correct me, or if I am oversimplifying I apologeize.
Besides it being absolutely wrong, you cannot boil the world down into Oppressed-Oppressor dynamics, this integration of all progressive causes leads to such ironies and conflicts within the Progressive-Left as the aftermath of what happened on New Years Eve in Cologne. Ostensibly a glaring example of Rape culture, a horrendous attack against women.... But the attackers were entirely, or almost entirely, from another member of the Intersectional Solidarity group, Arab and North African men of predominately Islamic faith. What do you do? Well they elected largely to side with the men, shunt the women off and revealed the obvious gaping hole here. If conflict happens between any of the groups within the Intersectional collective, the whole kit and kaboodle falls apart. Or for example the conflict over the Oscars being "too white." even though plenty of Latino actors won, the Latino community suddenly found once Black America isn't benefiting or in the spotlight they go from "Fellow Person of Color," to "White devil," on a dime.
The inherent instability and incoherentness of the "movement," and its attempt to build a meta-narrative of "Cishet white male privileged shitlord against all the world," collapses and will continue to dissolve away as the hell mouth of Intersectionality binds them onto a sinking ship of disjointed movements with no clear goals and power cliques vying for dominance and to merely use other groups for extra labor in the "Progressive-Stack." As it appears, if 2015 was a year of Progressive triumph, 2016 is a year of Progressive decline. Simply put its an unstable coalition much like the money worshiping big business and Christian Conservatives were never destined to work out.
Intersectionality allows many movements to theoretically work together, but in this "I scratch your back, you scratch mine," set up it first requires an epistemology that is correct (It isn't) but also requires that all the constituent parts never bump into each other, only the evil "oppressor." But the constituent parts continue to run headlong into one another. Revealing Feminists who apologize for rape culture, PoC who don't care when PoC win an Oscar, Trans* activists that hate on a notable Trans* people, ect ect as contradictions build and the public looks on with ever greater skepticism.