Why using an audio chip to get a triangle wave?
Short answer - Because you can.
Most of the time we don't want an audio amp to oscillate but they have a natural tendency to do so. The 'designer' probably had a few of these chips lying around and decided they would do the job.
Strictly speaking it doesn't produce a triangular wave but two exponentials as it charges and discharges the capacitor.
One possible advantage of the audio amp chip is its drive capacity so a low value resistor could be used to charge and discharge the capacitor. The 'triangular' signal is taken across the capacitor so a high impedance drive would be problematic.
How does this particular oscillator work?
Regardless of the pin names the 820 internal circuit shows pin 2 and 3 as effectively the inverting and non-inverting inputs but unlike the op amp the output is internally biased to half the supply (making Vp1 = 2.5) so that it can give maximum swing. This allows the 820 to use a single supply without additional offset resistors.
The circuits shown compares the relaxation oscillator version and audio amplifier TBA820M pinout.
The output of the relaxation oscillator has two feedback loops. The resistors (R2 and R3) form a (switched) reference voltage at the 'non-inverting' input to compare the voltage across the capacitor at the other input.
The capacitor charges up to the more positive value (output is high). Once reached the output switches low. The capacitor then discharges to the lower level. Once reached the output switches high and so on.
What looks odd is that we don't use the normal output (squarewave) but take the signal from across the capacitor.
How are the DC offset and triangle signal added?
The two 47nF capacitors separate the triangle waveform from any DC they may have (2.5V). These are coupling capacitors.
The potentiometers, P2 and P1, are connected by 10k resistors to the other side of the 47nF capacitor plates. The DC level of the wipers is then added back to the triangular waveform.