Why this date? The Talmud explains:
Said Rabbi Elazar in the name of Rabbi Oshaya: Since most of the rains of the year have passed, and most of the season remains. (Rosh Hashana 14a)
That is, Tu Bishvat is deliberately a midwinter event. The winter is Israel's rainy season, and Tu Bishvat falls at that delicate point when most of the rains are behind us, though most of the winter is still ahead of us.
What does this have to do with tithing fruits?
Simple. If most of the year's rain has already fallen, then the halacha assumes that any fruits which are already on the trees have grown on the current season's rain, and thus belong to the outgoing cycle of tithes. Any new fruits to grow will, by presumption, be nourished primarily by next season's rains and thus belong to the next cycle of tithes.
But wait! What's the difference between a fruit which grows on Shvat 14 and one which grows on Shvat 15? They were both watered from the same season's rains!
The stupid answer: There is no fruit growing in Shvat! At least not in ancient times. Fruit trees blossom in the spring and bear fruit in the summer. Hence "Hashkedia porachat" - the almond tree is the first tree to blossom, even before Tu Bishvat. But it doesn't bear fruit yet!
Tu Bishvat was chosen as the cutoff date for tithes because it is firmly between last year's and this year's growing seasons for fruit. All fruit clearly belongs to one side or the other.
Well, almost. Even in Talmudic times, it was clear that some fruit didn't fit the rule. Etrogim, in particular, can remain on the tree from one year to the next, and grow throughout the winter. The Talmud discusses their status, and ultimately the halacha is that Tu Bishvat doesn't apply to etrogim. They are tithed based on the date they're picked (not grown), and their cutoff date is Rosh Hashana, not Tu Bishvat.
So we eat dried fruit on Tu Bishvat (a much later custom - Tu Bishvat was not treated as a festival until at least medieval times) because fresh fruit is (generally) not available. Modern Zionists, trying to recruit Tu Bishvat to their land-building efforts, started planting trees on Tu Bishvat, despite the fact that it's a lousy time of year to plant trees.
But you don't have to do anything if you don't want to. Unless you have fruit trees in the land of Israel, in which case you should do your taxes!
Update: Judith Weiss of Kesher Talk offers more great Tu Bishvat links.