I can tell you what has worked best for me… I retrain TBs from the track. They obviously don’t come with factory installed buttons for learning to use their back, so you have to teach them to.
First, I teach them to accept contact. That can be done many ways, but what has worked best for me is to teach them to lunge, and then lunge in side-reins. The side-reins are judicial and fair; they always have the best timing, and are more forgiving than any hand. I keep the side-reins on a fairly loose setting - loose enough that it does not have contact with their mouth unless they engage it by stiffening or inverting. This is done in hand until they learn to lunge, and what I will do is instead attaching the side-reins near the girth area, I attach them from the bottom of the girth with the girth ring. I pull the side-reins through a neck-strap (or stirrup leather, or whatever) so that the full weight of the side-reins don’t jangle on their mouths when they’re moving. This is simply to teach them to accept and give to the contact - I don’t expect them to work in side-reins once that basic understanding of contact is there. They catch on pretty quickly, so I usually will do two weeks of lunging sessions that are kept short and sweet with side-reins and then put the side-reins away for the rest of the year.
To get a young horse, or TB, or whatever - to use their back, I start on the ground. Like MysticOakRanch said, this does not happen overnight. Most horses don’t even know how to leg yield, or to do lateral work… so this is the most important part.
In hand, I ask them to move over by touching their barrel with my hand, right where your foot would be. I ask them “over”, and nudge until there is a response. Release immediately, reward. Try again. At first, you are looking at the horse doing something like weird turn on the forehands, but do this every day for 5-10m before you get on (and if you’re ambitious, 5-10m after you get off) and very soon you will have a horse that understands that “over” means move his hind end away from the pressure.
Once he understands that, I graduate to asking him to walk along-side me, and ask him to move “over” every couple of strides. I am walking alongside him, so visually can see that hind end; ask him ‘over’, nudge - the timing is important. You want to ask him just about when you see the inside hind begin to engage in push-off - a split second before is ideal, but it takes time to develop perfect timing. Each stride of the inside hind pushoff, ask him to push off and push inside hind OVER, think about having his inside hind end cross over his tracks and almost where his outside hind would be tracking. THEN he is using his back!
If you do that for 5-10m a day, walking alongside and asking him to leg-yield (as that is what it is now, a leg yield!) towards the wall, you will eventually see his back start to engage, as he has to push off with his hind end to oblige your request. If you really work on getting a big, active push from that inside hind (remember to change direction) every day before you get on, you will be halfway there as he will already be using his back before you have your feet in the stirrups.
Doing a leg yield correctly will loosen the horse’s back, and engage it - they have to step under themselves to move their HQ away, which is why I always hand-walk and ask them to do a lot of moving over and leg yields before I get on. You want him to move over without speeding up - this will take time (rhythm).
Once you are certain he understands that, you can graduate to asking it undersaddle. In the walk, ask him to do lots of figures, changes of bend inside and out, TOH/TOF. Lots of greenies will collapse their inside shoulder when you ask them to leg-yield, so you need to keep that in mind once you start asking to LY under saddle.
Under saddle, remember to do lots of transitions. I like to do a four loop serpentine and halt or walk in the center of each loop, resume trot, change bend, halt or walk again.
I find that if I keep my contact light, but my legs strong, they start to offer reaching for the contact themselves. It won’t happen in a day.
Generally, study the training scale and have a good set of eyes on the ground for you.
When a back is engaged… it feels different. It is different for every horse, but generally, it feels like “the pieces fell together” - light contact, uphill swing of the shoulder, and you should feel a “push” from behind that almost pushes you out of the saddle. To me it reminds me of a coil, or bedspring - coiling in and out. Generally the trot will feel “bigger” without being faster, and the canter will feel more uphill. The walk will feel purposeful, and YOU should feel like you are dictating every stride with coiling, elegant purpose.