How is the sound produced in the sousaphone?

The sound of any brass instrument starts with buzzing your lips into a mouthpiece, which agitates the column of air inside the tube that makes up the instrument, causing vibrations in that air column which travel through the outside air as sound waves.

By buzzing your lips faster, you can force multiple waves inside the tube, so that (for example) two equal waves are occupying the tube instead of just one. Since this means twice as many waves are produced in the same amount of time, you have a higher frequency--thus a higher pitch (an octave higher, to be exact). There is a mathematical formula called the harmonic series that can plot the different frequencies (notes) that can be produced by this "overblowing" method.

The other way you change pitch on a brass instrument is by changing the length of the tube--shorter air column, higher pitch; longer air column, lower pitch. Trombones do this the easy way--they have a slide which, the further you push it out, the longer the tube, thus the lower the pitch. This wouldn't be very practical for a tuba or sousaphone (though it would be fun to watch), so instead they have valves (usually three on a sousaphone) that redirect the airflow through extra bits of tubing to make the air column longer and thus lower the pitch.

Between the two--changing the speed of lip-buzzing, and using valves (or a slide) to change the length of the tube--the brass instruments are able to cover all the notes of the chromatic scale.