Exercise execution is similar to form, it is how you lift weights, the definition would be the way we perform a movement or particular exercise and the stress that muscle is receiving.
It is all about getting the body in the correct position, safely and effectively and then performing a movement while putting the maximum amount of stress on the target muscle.
Now this shouldn’t be at the expense of progressive overload (increasing the weight you are lifting over time), you should always perform a movement safely and with correct form but still while aiming to increase strength in the gym, they should work hand in hand.
Any program, with the aim of gaining muscle, should be primarily based off gaining strength on multiple compound and muscle specific lifts.
The problem comes because often as people increase the weight they are lifting they allow their form to deteriorate and as a result it often leads to injury.
To add to this, the added weight isn’t being targeted properly so you don’t see the increase in muscle size as a result.
Exercise execution is incredibly important because it allows you to comfortably target the prime mover in a movement (the main working muscle).
If you're lifting a heavy weight but not effectively targeting the right muscle then you're not going to see the subsequent muscle growth from that. Therefore, the increase in weight isn't actually allowing you to progress and is only making you more prone to injury.
The only problem with focusing on exercise execution comes when one tries too hard to get the perfect contraction that this actually becomes counter productive. The added stress on focusing on the contraction of the muscle actually inhibits the maximal weight you can lift.
In other words, you could have lifted a heavier weight, with a great muscle contraction, for more reps but instead have exerted more energy than necessary trying to target the working muscle.
In summary, weight lifting really is a skill, execution of repetitions should be done in harmony with increasing the weight you are lifting. You should not be focusing so hard on the “squeeze” that you actually inhibit the force you are able to produce.
But, you should be focusing on lifting heavy weights, safely and effectively while feeling most of the tension of the movement in the primary muscle you are targeting.
That is ultimately the blueprint to seeing the results you desire when it comes to gaining muscle and for longevity of training.