When you're an athlete at any level, you hear people telling you to "listen to your body." I've been trying to do that and you know what? I'm done with it. The only time I ever hear anything from Mr. body is when Mr. body has something to complain about.
Geez! Just suck it up, Mr. body! I can make your life easy or I can make your life difficult, so if you don't have anything positive to say, just shut up and do as I ask...
Ok, now that I've gotten that off my chest, I do have some things to say about "feeling" the body, rather than "listening" to it.
Hey! Stop touching yourself! That's not what I mean by "feeling" the body... What's wrong with you?
What I really mean, I guess, is "sensing" the body... Ok, poor choice of words to start, perhaps. My bad.
Anyway, it seems to me that your body already has to be screaming before you can hear it anyway (i.e., you have to be injured or hurting), so listening may not make sense. But you can "sense" the body at any time. Give it a try. What does your mouse feel like... No, leave the rodent alone! The one attached to your computer... Great. What is your thigh feeling? Pressure from the chair you're sitting on? the hem of your shorts? How about the ball of your left foot? Don't just feel on the surface, though... Feel deep down... In the middle of your thigh muscle... The bone even... Feel anything?
Anyway, you get the picture. Now, I recommend a little practice. Try rotating your focus around to different parts of your body, perhaps in some systematic pattern. Feeling on the finest scale possible the sensations (if there are any) from the tips of your toes and fingers to the top of your scalp. Do you find anything is tense that you hadn't previously realized? I'll bet you do. Are your shoulders tense? Why? Can you relax them?
When you become adept at sensing the body, you can avoid wasting energy by remaining 'tight' when you are running or cycling. You can sense your body's position better. Sense how your foot is REALLY striking the ground. Sense the degree to which you are leaning. Sense whether you're form is a little off or whether something you are doing may be putting undue stress on a muscle or joint. Sense whether you're pushing off too hard with your calf muscles, perhaps straining your achilles tendon a bit, or whether your tensing your shin to dorsiflex your foot when you swing your leg forward. Are your hips level? Are you letting them flop lazily? Is your posture good? You can sense all these things with some practice, and learn to correct your form as your get tired (or lazy).
Once you can sense your body well, you can identify where some of your form problems might be. You can learn to make adjustments in your form on the fly to prevent injuries or to squelch certain types of discomfort before they become problematic. But here's the rub: you need to calibrate your senses like any fine instrument.
Sensing certain aspects of your form, particularly how you are moving, isn't easy if you can't also see what is going on. So, practice sensing body position and movement, but also get feedback from others or make some videos of yourself running to verify what your senses are telling you. I did this once when I was trying to learn how to extend my hips as my leg swung back during each stride, and saw that I was exaggerating the motion hilariously, and looked like I was going to pull something in short order (and probably would have). I needed to fine tune my sense of my form. With some practice, I learned how to sense this aspect of my form with greater subtlety, and regularly now check in with my legs and hips to verify my form and make often subtle corrections, particularly late in runs when fatigue makes it difficult to maintain good form.
A good way to just get accustomed to feeling the nuanced sensations of your body is to learn to do a body scan. A body scan is a form of meditation where you just systematically focus on different parts of your body as I described above. At first you may suck at it. After a few tries, you begin to get a feel for it, though. You started out sitting or laying down with your eyes closed in a quiet place, and over time you become capable of sensing any part of the body under a variety of conditions, even during a race or while you're out on a long bike ride.
So, try sensing your body, now and then and being proactive when you can tell something is not working ideally, because if you're trying to "listen" to it, all you'll ever hear is complaints and by then it's often too late to do anything but stop... And who wants to do that?