I'm not completely new to analysis, but I'm an engineer -- very applied, not very theoretical -- looking into self-studying pure mathematics. I've recently stumbled upon Henstock–Kurzweil integrals; many internet articles state that it's a very intuitive idea, but I simply can't wrap my head around its intuitive meaning as successfully as I can do it for the Riemann integral. The Wikipedia article seems to be pretty well-written, but I probably need a simpler stated approach/definition, because I'm only starting to get into all of these things.
Can someone give me their own explanation of what the Henstock-Kurzweil is, or perhaps a good resource?