I understand that when trying to explode in situations such as sprinting, striking, and jumping, ideally you want to be able to quickly alternate between bouts of muscular contraction and relaxation. To be more specific, they say that you first want to explosively contract the muscles of limbs your are intentionally moving at the utmost beginning of either launching a strike or pushing off the ground and swinging your arms (depending whether you are striking, jumping, or sprinting) in order to initiate a high speed of muscular force. Then, right away those same muscles should relax to reduce muscle stiffness during a given motion, thereby letting the limb pick up speed through it's motion.
Finally, to repeat the same movement or quickly do another kind of movement you repeat the same sequence of phases of muscular contraction and relaxation that you did with the first movement. I get all of this; however what I don't get is whether or not the same idea of alternating between contracting and relaxing applies to explosively lift something that is of moderate to heavy weight. I mean, how can you relax your muscles when you are moving a moderate to heavy weight in order to lift fast?
Usually you will told you are being stiff when you are using antagonists or stabilizers to brake the movement to over control it. Basically the wrong muscles are being engaged. For example, with a punch, you should have your lats contracting isometrically, you tricep contracting and your bicep relaxed as the punch is thrown, then the tricep relaxes and the biceps and lats contract when you snap it back. Your shoulder should be relaxed through the whole punch. You shouldn't be slowing down ahead of full extension by contracting other muscles.
It will be "stiff" if you are braking with your tricep or other muscles in your arm as it goes out to "control" the punch, or to slow it down before it reaches full extension. This is one of the problems of punching the air and holding it like you do in a karate class. The punch should be thrown and retracted, not stopped, if just encourages stiffness by encouraging a de-acceleration phase in the punch.
So in the case of a weight, if you are over guiding it, or slowing toward the end of the eccentric you are being stiff and not explosive. What I was told when I was starting out was line everything up and throw the weight. If you are having trouble in part of the lift find the right assistance lift to focus on the part of the lift you are having trouble with so you can just line up and throw the weight. Don't try and force it part way through the lift.
Okay, that makes a lot more sense. However, I thought that you were supposed to tense all the muscles of your striking limb as well as you entire core at impact, even if you do multiple strikes continuously. Also, when you said "it will be "stiff" if you are braking with your tricep..." did you mean to type bicep and not tricep?
While you do want to be stiff at the point of impact, you shouldn't be focusing on it. From my own training, and for most people I've trained they start out stiff and push the punch, as once their fist is clenched they have muscle tension through the entire punch. They also decelerate as they come to the end of the punch as this is natural to protect the shoulder. This is slow, doesn't land against a trained fighter, and because it is slow it also doesn't do as much in the way of damage. It transfer more total energy during the punch, and if you ever get into brick breaking it works quit well, but it isn't a proper punch for hurting someone.
Once I get them to relax, they unclench the fist and relax the whole arm and "slap" with a relaxed fist. This is quick, but it isn't going to hurt anybody either because an unclenched hand absorbs the impact rather than transferring the force. If they clench just the fist throughout it will be better, but basically you get the energy of the fist transferred to the point of impact only. This will hurt, but it isn't full power. Full power comes when you contract the the arm, the core and drive with your leg and hip all at once to put the entire force of your body behind the punch. Basically everything needs to be rigid to transfer the power. But it needs to be for a small, small fraction of time, instantaneous if you can manage it.
Basically what I found fixed this quickly for most people was heavy bag and double end bag, and feeling the punch more like a boxer does. You will never land twice on a double end bag if your punch is too stiff and you push it, and you will feel it right away on the heavy bag if you are slapping without tensing as the bag won't move. If you just contract the fist properly, you can get a punch that will hurt, but you are just transferring the force of the acceleration of your fist. If you get that moment of rigidity in the arm, you get the force of your entire body. By and large it made me unpopular with most of my karate club as there is a rivalry with boxing, but whatever works.
There is no equivalent in the olympic lifts to that point of impact though. While you need to have your support muscles contract isometrically, they pretty much remain contracted through the lift. You aren't transferring energy in a momentary impulse, you are accelerating the weight with a constant force throughout the movement (if you can).
This was a problem I had bouncing back and forth between throwing and martial arts. I developed the technique for transferring at an impulse and ended up only adding force momentarily at the beginning or end of each movement rather than accelerating through the whole movement. A smooth pull where you accelerate throughout is much better for moving a weight, even quickly, then to try and just put the force into a momentary impulse. By applying force through the entire movement you transfer more total energy to the bar and can accelerate more weight faster. By transferring everything you have in your body in one instant in time, you cause more injury with a punch.
Antagonists don't remain contracted during an olympic lift or a throw. Ideally they won't be used at all. This should be all concentric, and a constant contraction. There is a point of rigidity in a punch, not in an olympic lift or a throw.
By supporting muscles, I was referring to whatever part of your body you need to remain stiff to transfer power during that phase of the lift or throw. It won't be a prime mover or a stabilizer, and especially not an antagonist. For example, I don't roll my upper back during your first pull of a snatch or clean, so my upper back needs to be statically contracted, and was told "look up and pull your shoulders back" as advice to encourage this.
I think that is as far as I can explain via text. You probably should get a coach IRL to take a look at your movement pattern. If you do the movement, a good coach can tell you right away what you are doing wrong, then you can get feed back and a better understanding than from anything I can post.