I suspect it's due a number of factors. Yes, body armor mitigates the fear/cost of being hit to some degree. On the other hand, I'd rather be hit by an 88 mph fastball than a 98 mph one. In that vein, some of it could be more velocity meaning less reaction time.
I wondered how the increase looks if you measure the HBP rate by pitch instead of by PA. B-R only has pitch totals going back to 1999, but just over that time, you can see that about 25% of the increase can be explained by increases in pitches seen per PA.
From a correlation standpoint, of the various PA outcomes, HBP/PA correlates most strongly, BY FAR, with the strikeout rate. More strikeouts = more HBP. It actually has a negative correlation with OBP, though small positive correlation with walks. The correlation with SLG or ISO is also moderately small -- about half of SO. This suggests to me that the biggest single factor is how nasty pitchers' stuff is. The faster the velo and the less predictable the movement, the harder it is for the batter to get out of the way. I'm sure batter willingness to take a pitch (due to body armor and valuing OBP) has also played into it, but I'd guess those are lesser factors.
In any event, while it has certainly increased over time, there have been noted periods of big change. The increase in recent years is part of a much longer trend.