The way I understand it, this is more of an issue on upright pianos than grands. It has to do with the construction of the action, the system of levers and hammers that transfer the energy of striking the key into the energy of the hammer striking the note.
If you own an upright piano at home, I highly recommend dismantling the front panel (ask yer parents, if necessary), and removing a key or two to see how it works (put it back when you're done playing). The key has a screw at the back called a capstan, and on the bottom of the key, there's a groove where it sits on a pin. When you press the key down, the pin stays put and the capstan goes up. The capstan connects to the rest of the action (the pieces of which all have clever, less memorable names).
Ignoring the middle part of the action which I am not qualified to explain. The big difference between an upright and a grand is the direction of travel of the hammers. On a grand, the hammers strike upward and gravity can pull them away to allow the strings to vibrate freely.
Not so on an upright. The hammers move horizontally. So there's little assistance from gravity to get the stupid thing back out of the way. There's usually a spring system to catch the backswing of the hammer, but it's a piece that can stop functioning correctly on unkempt uprights. That means if you're not careful, you can double-strike the string.
One of the dangers of practicing too much on an out-of-repair piano is that you teach your fingers not to trust the catchback system. And you try to control the hammers with your fingers, pressing and holding (with pressure) where finesse is required.
You should be pressing the keys so as to launch the hammer against the string and follow through with the movement until the key hits the bottom (there should be a pad, but call it "to the wood"). One firm strike and then your finger can just stay there until releasing, when the dampers will reengage.
The catchback system and the action (height adjustment of each capstan) are the pieces that need to work correctly. And if they do, then your fingers need to work with them, rather than against them. The capstan heights affect the "play" between the key and the rest of the action. On an out-of-repair piano there can be quite a bit of play between the touch of the key and the point where it engages the action. Even worse, the play may vary from key to key, so you have to alter your attack of each note to try to get a consistent volume. This contributes to much of the slappy sound of the old western saloon piano meme, as there can be a extra percussive noise if there's both play at the capstan and worn-out pads at the bottom of the action where the capstan strikes. So much for the piano mechanisms.
As to how this knowledge affects playing, the act of pressing a key should be more of a lunge rather than a slap. Imagine you're poking your stupid brother in the shoulder, telling him he's wrong about something. Not hard enough to hurt, but hard enough that he knows he's getting poked in the shoulder. :)
The part your teacher mentioned about releasing the tension is also very true. After the hammer is launched toward the string, it has a moment of free movement. The whole hammer assembly disconnects from the part that gives it force and at this moment (just before it strikes the string), you cannot affect it's velocity anymore. The hammer needs this moment of freefall in order to engage the catchback system at all. If you're pushing the keys like a bad piano might train you, the hammer doesn't experience this freefall moment, and you can continue to apply pressure right up to the string. This is a problem I had myself. (My Russian piano teacher would say: "Joshua, why do you pet the cat?" "Why do you tickle the keys?" Years later, I figured out what she meant.) In effect, your fingers try to hold the key with force, to replicate the catchback system. As programmers say: don't re=write standard library functions.
The difference is more about speed than force, I think. Not in the sense of the physics, but more poetically. Think about pressing the key in one swift movement. Remember that about halfway through, the hammer launches off into freefall. So you only need to apply pressure at the very beginning. One swift stroke.
If you can manage to practice on a grand regularly, it will help to retrain your fingers. You simply cannot control the hammers in the same way. The extra unnecessary pressure will give no haptic feedback, and this may help to loosen the habit.