...so why is this year's inauguration on Monday the 21st (the Martin Luther King holiday), and not the day before?
It's for the same reason that Sally tells Harry they don't make Sunday underpants: because of God.
Which is to say, because it's Sunday.
I had thought that this was constitutional, but it's not: it's just a tradition. You can read the brief version here. This happens -- roughly -- once every 28 years (there are disruptions for various reasons). It happened with Reagan in January of 1985, Ike in January of 1955, and Wilson in March of 1915 (remember what I said about disruptions?). The current tradition (which was done with all three of those earlier gentlemen) is to have a private swearing-in on the actual, uh, inauguration day, and then a big public hoopla on the following day, including a do-over of the swearing-in ceremony. (Which happens: they had to do it last time too, after all, albeit for different reasons.)
It would be awkward if there was actually a new President, and not a second term, but coincidentally that's only happened twice, and not for well over a century: the two times were Hayes in 1877 and Taylor in 1849. (The first of these lead to the fanciful notion that David Rice Atchison was actually acting President for a day.) The other four times -- the three Twentieth Century ones, plus the first occurrence, with Monroe in 1821 -- all happened to be, as this one is, a reinauguration. It certainly hasn't come up since the Presidency became what it is now, i.e. the sort of office where you need to know at every moment who is in it. (In 1848, a day here or there wasn't that big a deal.)
Perhaps that will happen next time, which will be (unless unforeseen disruptions occur) on Sunday, January 20, 2041. (If not then, then next up is Sunday, January 20, 2069, and then Sunday, January 20, 2097.)
(To say it's ridiculous to speculate about parties and presidents 28 years into the future is to wildly understate the matter. (Will there even be the same two parties? Will our constitutional structure have changed? Who knows.) Nevertheless, I can't help but pointing out that since WW2, with one exception, the parties have alternated eight-year stints. (The exception is that under this scheme Carter should have won the 1980 election, so that that particular sixteen year stint was divided 4/12 instead of 8/8; change that outcome, and no other, and the pattern holds precisely.) I consider it extremely unlikely that this pattern will hold for twenty-eight more years. (Indeed, I hope it doesn't: a Republican administration in 2016 would be a disaster for the Republic.) But if it happened to, then January 20, 2041 would indeed see the inauguration of a new President, and not a reinauguration.)
Normally I would natter on about this -- intersecting, as it does, various of my obsessions -- but I don't have to, because this marvelous post I found in looking into it says everything I'd want to say about it and more, including remarkably complete histories of each of the previous incidents, explanations of both disruptions of the 28-year-cycle, etc. If you've read this far, and you're still interested, click on over.