I need assistance in trying to understand this schematic. I think current might be oscillating back and fourth and that signal is being amplified by the op amp, but really I have no idea.
At first glance it looks like the 741 opamp is acting as an integrator, which uses a gate or trigger signal from the pushbutton to create a ramped output. R2 & R3 likely change the RC time constant for different envelope characteristics, and it looks like R4 is possibly a threshold control for the trigger input. The LM386 is acting as an amplifier to drive the speaker output.
I'm not too familiar with this particular circuit, so maybe someone else can chime in on the nuts & bolts of how it works. This would be a perfect candidate to simulate in LTSpice!
That circuit looks like it is based on the Radio Shack Engineer's Min-Notebook series.
Most of them (maybe all, I'm not sure) are available to read on archive.org
Here is the book for the synthesizer project -- it is on page 39 and looks very similar to the page posted above. The earlier parts of that book may help explain the layout for it.