As I understand intersectionality, it is about identity politics and recognising that we all have multiple causes to support, to feel aggrieved about, to determine that other people are better off than we are.
At its core, it is nothing more than sociological babble for ‘we are each individual, with some things in common with the next person and some things different’ but it seeks to highlight that we belong to groups of people with something different than the next person. It is therefore a useful tool to claim grievances as a class. (This group/class identification goes back to even before Marxism and Friedrich Engles’ influences on group – intersectionality is not such a new concept even if the word (in English) is new.)
There can then be a hierarchy of ‘privilege’ and ‘oppression’ defined for each group and the more such groups you can belong to, the more grievances you may claim, or the more you may be blamed for the oppression leading to those grievances. For those who feel put-upon (don’t we all, at some point), or are taught that they are part of an oppressed class (e.g. ‘Gender Studies’ classes) they will naturally seek to identify with any group which it is politically expedient to determine as the under-dog. Meanwhile, regardless of any evidence such as statistical data and legislation, those in classes determined to have privilege (in the USA, this is primarily those who have pale skin and those who are male) are to be reviled for having privilege.
On that last note: