How does this spectrometer sensor output its data and how to process it?
The module has a linear CMOS sensor to detect brightness of different wavelengths. The readout is a 5V analog signal level for each pixel with a sync pulse.
Unfortunately the process needs a minimum clock of 200 kHz to set the pixel scanning rate, so the Arduino would need to provide at least 200 kHz clock and sample analog values in sync with the clock.
The ATMega328P ADC has a maximum sampling rate of 15 kHz so it is not possible with it.