Why does the frequency of a wave remain constant?

The frequency must remain constant to avoid a discontinuity at the boundary.

The easiest way to see this is to consider 2 ropes of different linear densities - e.g. a thin rope and a thick rope - joined in series.

If you shake one end at a frequency f, then (transverse) waves will travel along the joined ropes. The waves travel slower along the thicker rope than the thin rope.

At the junction between the ropes (and to either side of the junction) the frequency must still be f - it wasn't the rope would have to split due to adjacent points having different frequencies.

The same is true for any wave - you can't have a sudden jump in the electric field of an EM wave for example - the electric field can only vary continuously, with no discontinuities.

As a consequence of remaining constant, wavelength and speed change proportionately (e.g. if speed doubles, wavelength doubles).

The assumptions under the statement are that A. the oscillation count in a wave is conserved and B. the passage of time is universal and uniform. Since the frequency of a wave is the count of oscillations measured within a given time interval by a stationary observer, it remains the same anywhere the wave can reach. On the other hand, the wavelength is how far a wave travels from one oscillation to the next thus depends on how fast the wave is traveling in the medium. There are cases we can find either A or B is violated, for example, in a nonlinear medium or a gravitational field, respectively.