Why harmonic distortion and not at other frequencies?
A sinewave of 1 kHz only contains one frequency: 1 kHz. Let's describe that mathematically:

\$x = sin(2  \pi  f t) \$

Where \$f\$ is the 1 kHz, \$t\$ equals time and \$X\$ is the sinusoidal signal.

If an amplifier is ideal then it would only amplify the signal, i.e. increase the amplitude:

\$y = A  x  = A  sin(2  \pi  f t)\$

Note how that still has is just a sinewave, only the amplitude (value of the minimums and maximums) have changed.

But that's a linear amplifier, it will not introduce harmonics.

Now what if the amplifier distorts.

Do you remember the Taylor series? It is a way to express any function in the form of a polynomial like this:

\$y = A x + B x^2 + C x^3 ...\$

What I wrote there is the Taylor expansion that describes the behavior of an amplifier with distortion.

If you fill in \$x = sin(2  \pi  f t) \$ you will get \$sin(2  \pi  f t) \$, \$x = sin^2(2  \pi  f t) \$ and \$x = sin^3(2  \pi  f t) \$ terms and these are the harmonic frequencies.

Note that there is no way to get terms other than \$x = sin^n(2  \pi  f t) \$ making it impossible to get frequencies that are not a multiple of the base frequency of \$x\$.

Bonus question:

What would be needed to get other (non-harmonic) frequencies?

With a sinewave as input, there is no way. But if we combine two or more sinewaves of different frequencies, then we can get intermodulation products. For example, make \$x\$ a signal consisting of a 1 kHz (\$f_1\$) and a 200 Hz (\$f_2\$)  tone:

\$x = sin(2  \pi  f_1 t) + sin(2  \pi  f_2 t)\$

Then at the output of a distorting amplifier we will find sum and difference frequencies so we would get:


200 Hz
400 Hz ( 2 x 200 Hz, the 2nd harmonic of \$f_2\$)
600 Hz ( 3 x 200 Hz, the 3rd harmonic of \$f_2\$)
800 Hz (1 kHz - 200 Hz)
1 kHz
1.2 kHz ( 1 kHz + 200 Hz)
1.4 kHz ( 1 kHz + 2 x 200 Hz)
etc etc


note how they're all 200 Hz (\$f_2\$) apart!

How many frequency components are present depends on how much the amplifier distorts and the amplitudes of the signals.