The number of cycles can be calculated by counting the peaks. In this instance, there are ten positive peaks in one-tenth of a second. This means that there were ten complete cycles of the waveform in that time period.
A single cycle of the sine wave represents one back-and-forth movement. When the tone is sent to a speaker, the positive peak will move the speaker cone in one direction and the negative peak will move the cone in the opposite direction. If there are ten cycles in one-tenth of a second, multiplying by 10 tells us that there would be 100 complete cycles in one full second. Since this is a 100 Hz tone, the result obtained in this way matches our expectation.