"Intersectionality" starts from a common sense notion, namely that oppression can take many different forms, i.e., sexism, racism, homophobia, transphobia, classism, xenophobia, etc., and therefore the experiences and perspectives of a rich, black, gay male will be different than a poor, white, disabled lesbian, for example. Since most of the social justice movement is comprised of white, cis-gendered, heterosexual females, and usually congregates in places of privilege (college campuses and Tumblr), the idea behind intersectionality is that more of an effort should be made to give voice to the underrepresented and oppressed. This is good.
However, as it is practiced by most of the young people in the modern social justice movement, intersectionality usually means that the person who can claim the most "intersections" is automatically afforded the most adulation and given the loudest megaphone during any conversation, regardless of whether their ideas warrant it.
This is especially problematic because it makes the social justice movement a magnet for narcissists who like getting preferential access to a soapbox without having to rigorously defend any of their ideas.
Also, since white, cis-gendered, heterosexual females (again, the bread and butter of the SJ movement) are immediately outclassed as soon as a person of color enters the room, this has led bored, privileged millennials to actively seek out ways in which they can claim oppression and therefore jockey for a position at the SJ table. Trans-genderism, trans-racialism, trans-ablism, and trans-cognitive are now officially recognized ways for people who were born male, white, healthy or intelligent to appropriate the struggles of women, the non-white, the disabled, or the retarded.
Before you claim this is hyperbole, I once volunteered at a place where a person with two fully functional legs had to be pushed around in a wheelchair by other volunteers because she identified as a paraplegic in order to be on an ADA advisory and activist committee.