This is one of my favorite cards to pick apart because it provides such a dramatic diversity of impressions. Rather than ask the usual questions, I'm going to offer a few answers.
In an affirmational culture obsessed with the mantra "It's all good," it's too easy to see this card as simply meaning "Oh, you're healing." It does in fact suggest an old-fashioned blood-letting or leeching to let out the "evil spirits." But I think it's more complex than that.
Right off the top, what's a card in the unemotional suit of Swords doing so blatantly portraying upheavals of the heart, the usual province of Cups? Furthermore, why does it show such a dire state of affairs when the numeration of the card (Three) typically expresses growth, progress and opportunity? Finally, why are there four elemental emblems in a card intended to show "three-ness?"
At least part of the answer lies in the three swords as a curb on runaway emotions. They pierce the heart like push-pins holding a butterfly specimen to a mounting board. The image also reminds me of a bounty hunter relentlessly pursuing a fugitive criminal. Then there is the logical assumption of examining one's feelings under a microscope. (Never mind that the RWS version also looks like a Mr. Coffee machine on "drip.") The idea is that the rational mind holds sway over the vagaries of feeling. The Thoth title of "Sorrow" seems to convey the understanding that the heart doesn't much like to be told by the mind what it can and can't do.
In the system of Elemental Dignities, Air and Water are considered neutral and mutually supportive; I like to call them "complementary opposites." One provides what the other lacks. In this case, there is an overabundance of Air to dominate and suppress the tendency of Water to flow wherever it wants, without regard for boundaries. This is why I often see the 3 of Swords as suggesting a minor and temporary irritation rather than a total collapse of personal happiness. The three swords aren't dismembering the heart, they're like sutures holding it together through force of Will until it can mend itself! See, it's "all good" after all.