Well you must admit it makes for a good story. I mean, SIX wives? Very soap opera.
I think it's also maybe partly because he's kind of the first monarch where the sources give us a decent sense of him as an actual personality - and the same with a lot of the people around him. Go back any further and we know what kings and nobles did but it's a lot harder to get a sense of what they were really LIKE. However interesting the history, a TV drama or novel needs to be quite heavy on personalities, and if you were going to do a show about Edward I or something you'd pretty much have to make it up.
Henry VIII also seems to have something quite "modern" about him, in the way that he seems to have always been led by his heart (or other body parts) in how he conducted his relationships - always married for love or what he thought was love, ended the one arranged marriage because he didn't fancy her etc. Modern people can empathise; we can't so much with the medieval-royal pattern of private life (marry a stranger for diplomatic reasons and then stick with it til death no matter how it's worked out or how you feel). We're so led by our emotions these days; so was Henry VIII.
The visuals probably help, too (thanks, Holbein). There's just not much decent portraiture when you go further back. Having that strong mental image of someone before you even know anything else about them makes us more interested, I think. It's just how our minds work, or at least how modern minds have been trained to work, in our world where we're receiving and processing a lot of visual imagery all the time and are conditioned to judge things on appearance.
Maybe he also chimes somehow with a modern concern about tyranny (he really was a tyrant), bad government, a madman having the power to destroy the lives of everyone else etc. I always think he seems to have been a very similar personality to Donald Trump - the total narcissism, the seemingly genuine belief that he could create reality with his mind, that what he thinks must be true, that what he wants must be the will of God. Henry is scary.
And there's a lot of women in the story, which is rarely so true before or after. Probably appeals to female audiences.