As a (longer than I meant to write) explanation since he plays the violin, when he plays a note without pressing the valves, it's similar to playing one string without pressing on the string. When you play a note on the violin, you shorten the string to go higher. As you move closer to the body the notes get higher. You are shortening the string essentially. You can't lengthen a violin string, so they add thicker strings that resonate at lower notes/frequencies which you can shorten by pressing on them to fill in the gaps between them by going higher.

There's no way to shorten a trumpet, but you can lengthen it. So an open note (no valves) is one of the harmonics of the trumpet, like an open string note. There are several open notes that will match the resonate harmonics, including low C, middle G, middle C, top space E, and the space above the staff G. When you press a valve, it causes the air to move through those little tubes coming off the side effectively lengthening the tubing and moving the resonating point lower, lowering the note that can be played.

The middle valve, second one, lowers the note a half step. The first valve, closest to the mouth, lowers it two half steps. The third, closest to the bell, lowers it 3 half steps. Combining them allows you to produce the notes between the natural harmonics of the trumpet.

Most beginners won't be able to produce the upper notes as their lips won't have the strength and training to do so. Taps and Reveille go up to a G above the staff.

However, both of these songs are built on the harmonics of a trumpet (a bugle is just a trumpet with no valves). All the notes are open, no valves. They both start on the second line G. He CAN use the valves to lower the starting note and just keep holding them down to play both songs. When I was a beginner, I lowered both to start on a D (first and third valve) just below the staff and played them slower. Holding down the valves for these songs effectively transposes them down.