Boiling tofu causes the bean curd to exude moisture, so you are correct there is a structural difference between boiled and unboiled tofu. The heat tightens up the protein structure, hence affecting consistency. Boiling bean curd for braised and fried dishes is a long practised technique in Asia, partly for these reasons. Because of the structural change, boiled tofu can stand up better to more vigorous cooking methods like long simmering and frying.
The type of tofu (mainly its coagulating agents) are related to the efficacy of this method, but it absolutely applies to your typical firm supermarket brands (non-silken).
Ah, so the question is "Why does boiling the tofu change its texture?". Answer: it doesn't. Try doing this recipe without boiling the tofu, I think you will find it is pretty much the same. I think if the boiling serves any purpose at all it is to remove any last remnant of taste from the tofu. I don't mash tofu often, but when I do, I never boil it and I've never had a problem.
Firm tofu is basically just soft tofu that's been pressed and drained. If you went out and bought soft silken tofu then you could probably just mash it straight into your sauce; extra firm tofu is harder to mash simply because it has a much lower water content. But as Michael says, you could still do it, it would just be a little more "crumbly".
I'm not really sure that there's any interesting or sophisticated chemical reaction going on here; you're just getting it wet, and being wet makes it softer and easier to mash up. It's more or less the same thing that happens with pasta, except that tofu doesn't contain any starch and probably doesn't need the high temperatures, so I doubt you need boiling water for this; just soaking it would give you the same result and wouldn't hurt the flavour as much as boiling.