First off, intersectionality can look something like this:
It demonstrates that if you're a woman, the poorer you are the more likely you are to be obese as compared to men (there are problems with using the Body Mass Index as a measure, but we'll leave that for another occasion). What you have here then is the intersection of two durable patterns of social division - gender and class - that produces a certain health-related outcome. Digging into this to find out what's going on means analysing how relationships characterised by class and gender intermesh and are performed/lived in the context of everyday life. Typically, while intersectional analysis is used to understand how multiple oppressions interact and redouble one another, it can be used to think about advantage too. For example, how class, age and masculinity intersect can help explain UKIP's support among a particular demographic.
The problems - and controversy - starts when we begin talking about political agency. Intersectionality gets a bad name because it is associated with point scoring and privilege checking. It's a problem that plagued some of the early theorisation on this topic. i.e. Where is the common ground for intersectional politics? You as a straight black woman and your neighbour as a gay white man experience different forms of systematic discrimination at the hands of the same configurations of power, but how to come together? Simply on the basis of it being a nice idea, as Laclau and Mouffe suggested or is there something else? Well, there is something else and that is class - but not how it is typically rendered by post-structuralist/neo-pragmatist/identity politics "common sense" thinking.