The reason is that there is a valve between you and a very high-pressure main pipe. The pressure is not equal across this opening--- it drops to nearly atomospheric pressure, and the water dribbles out at a rate determined by the size. The opening allows a fixed amount of water per unit time through, determined by the pressure drop and Bernoulli's law.
When you cover the opening with your thumb you initially make the water dribble out slower, and the pressure rises closer to that of the main, so it will rise closer to the pressure provided by the water utility. The spray rate is again determined by Bernoulli's law, but now for the much higher pressure in the tap. You can go up all the way to the pressure in the main as you cover more of the opening, and if you cover all the opening, you get the main pressure on your thumb.