It’s a type of spectrometer, which is a device used to make measurement with light. In this instance you pick specific wavelength(s) of light which interacts with your blood differently depending on whether it’s oxygenated or not. You shine that through your finger, and then use a detector to check how much actually made it through (some of it was absorbed by your blood). You can use that value to calculate what the oxygen content of the blood is because you’ve already researched the relationship between blood oxygenation and transmission/absorbance of this particular wavelength of light.
The reason for differing interactions of light and blood in different oxygenation states is an entirely different but very interesting topic which I will not go into here. But the underlying principles make this type of measurement possible, and spectroscopy is a huge deal in chemistry.
An easy to understand, everyday example is how you can tell how much dye is in a cup of water by how dark it is, with you acting as the spectrometer. You know a darker or more intense color means more dye, and that red color means red dye, blue color means blue dye etc. So by observing the light that exits the cup with your eyes, you can make a rough judgement about how much dye is in there, and a statement about what color it is, importantly without having some magical method of counting dye molecules one by one.
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