Progression via added repetitions can stimulate the neuromuscular system in a number of ways. Among the neuromuscular demands posed by added reps include structural and energetic challenges. Structurally, added reps demand tissues to tolerate more exposures to tension and, possibly, change accordingly. Energetically, added reps demand muscles to manage energy-burn depending on the type if reps added. As such, the challenges and, therefore, adaptations resultant of added reps can heavily depend on the force of those reps.
Adding forceful reps
Adding forceful repetitions to your workload encourages structural/energetic adaptations for generating greater force. Nerves become larger to fire more muscle fibers. Contractile protein becomes thicker and stronger for more force. On the energy front, muscles store more fast-energy and fast-energy converters.
Adding fatigued/less-forceful repetitions
The less forceful/more-fatigued reps become, the greater the likelihood adaptations that lessen peak-force result. Nerves recede and recruit fewer fibers lessening force to conserve energy. Contractile protein can either become larger or smaller depending on how much energy is spent; less energy expenditure allows thickening while more energy expenditure tends toward atrophy. Fuel storage/conversion changes depend, again, on energy expenditure; less energy expenditure results in more energy stored while more expenditure results less storage but greater ability to access slow-energy from the blood.
Adding reps for greater strength
Adding repetitions can increase strength if said reps are sufficiently forceful. Within the range of sufficient force, there's variation how strengthening occurs. More forceful reps, while influential on protein synthesis, usually strengthen via larger nerves/greater-fast-energy potential. Lesser-force reps, within the spectrum, tend gradually toward thickening fibers via cellular-fluid/protein synthesis.
What to do?
Whether your goal is mass and strength or strength alone, push/pull weights as fast as possible. For mass and strength, add repetitions via increased sets and end the workout when force noticeably drops. For strength alone, add sets only if certain force output will meet/exceed the preceding set. Also remember that taking sets to failure greatly reduces force so avoid failure or reserve it for the last set.