The main benefit of short uphill sprints is probably neurological. As long as the grade is not so steep (about 14% grade is close to the upper limit) and the duration of each sprint bout not so long as to significantly alter or even impair stride mechanics, this represents as high a level of motor neuron recruitment as you can get in a running-specific context. The combination of speed and resistance produces an extremely high degree of both recruitment and "rate coding" in synergistic muscles. Believe it or not, when it comes to improvement, the speed is more influential than the resistance offered by the hill. But when speed and resistance are combined (and balanced in appropriate conjunction with level-surface sprinting or even gentle downgrade freewheeling), the ability to regulate force through proportional control is topped off.
Any exercise which recruits large motor neurons within major muscle groups can promote the release of growth hormone as long as nutritional requirements are met to supply amino acids. Squats and dead lifts do the same thing, but they obviously aren't sport-specific.
You might be feeling stronger as a result of a growth hormone boost, but it's just as likely you feel lighter and snappier because of the improved neural allocation (the benefits are noticed fairly quickly), because of having fortified your phosphocreatine system (which facilitates the warmup process) and because of the sheer variety in the routine.
To effect the best boost to the phosphocreatine system (and to improve proportional control at speed), work up to 3 sets of 5 or 6 of the short uphill sprints (you don't have to spend 12 seconds on the actual hill for this; a run-up of 2-3 seconds on a flat section to establish some speed before starting the hill is best for form retention, followed by about 8-9 seconds on the hill) with a slow jog down and a little extra time at the bottom, for a total recovery period of about a minute between reps within a set. This format allows for partial restoration of the phosphocreatine (starting with roughly 80% replenishment and dwindling with each successive rep) while still applying a functionally continual stress to the system. Take about 5-7 minutes (walking for a couple of minutes, then some jogging, then about another minute of walking) between sets to "almost completely" restore the system.
If you're a pure middle distance runner, you can use this session once a week during the last half of the off-season. For another supplementary addendum, tack on a single set of about 8 reps of short hill sprints (use about 3 reps to warm up to speed and get accustomed to the hill) following a medium-effort "tempo" type of workout on another day of the week (it's best to include some dynamic drills and several level-surface or gentle downslope high-speed buildups in your warmup prior to the "tempo" effort if you're going to approach top speed afterwards). So you can get two days per week which incorporate hill sprints - one is a full-fledged workout which is started completely fresh; the other is as an annex to a moderate-effort "high end" or "threshold" workout to briefly summon the full spectrum of motor neurons following mild to moderate fatigue of the more resistant oxidative muscle fibers.
In regards to oxygen uptake, sprinting for short distances with recovery periods, even when repeated dozens of times, can affect single stroke volume, but this has only a marginal effect on O2 uptake. The muscle fibers involved in full-speed uphill sprinting are of larger diameter than those used in sustained running and more space within these fibers is devoted to contractile structures designed for force production, leaving less room for mitochondria or blood vessel supply. In fact, when they are recruited, their contractile force is often great enough to constrict their own blood supply, meaning they are forced to rely on their own stores of anaerobic energy, which is one factor contributing to their quicker fatigue. Furthermore, the heart, being a "twitch" muscle rather than a "tetanic" muscle, requires a few minutes of continuous exercise to reach the kind of sustained stroke volume employed during fast continuous running. Such sustained delivery of oxygen is generally required for significant improvement to take place in O2 uptake.
Running-specific myocardial O2 demand (which stimulates the ventricular hypertrophy contributing to higher O2 consumption) is best created from segments of about 2 to 2.5 minutes of uphill (6%-8% grade) running at an effort similar to 5k race pace on the track. This does place a more continual pressure on the blood vessels which supply the race-specific (oxidative) muscle fibers, forcing a higher prolonged stroke volume than level-surface reps of the same duration would accomplish.