The triangle is an amplifier. In this case it is a POWER AMPLIFIER.
When you connect the "-" input to 0v, any slight voltage on the "+" input is going to make the output change from a LOW to a HIGH.
The first few milli volts do not alter the output, but after that it only takes a few more millivolts to change the output from a LOW to a HIGH.
In this case the amplifier has a gain of 200 and each millivolt rise on the input will produce 200mV rise on the output.
If the supply is 10v, the output can only rise to nearly 10v and thus an input of 50mV will provide a full swing.
But the input must have a capacitor so that on the AC signal will be passed to the 10k pot, allowing the output to fall back to zero when no signal is delivered.
The amplifier chip normally only has a gain of 20 and the 10u electrolytic increases the gain to 200.
The 250u only allows the signal to enter the speaker.
You will have to find out if the output of the op-amp sits at zero when both inputs are zero.
If the output sits at half-rail and the 250u is removed, current will flow into the speaker and shift the cone. When you try to move the cone further in the same direction, it will have already reached its maximum travel and thus the output will be distorted.
The 50n and 10R improve the quality of the sound by removing some of the highs.
