To introduce my situation: I'm developing a digital synthesizer in a form of a C++ library, working with low level APIs like WASAPI, ASIO, ALSA etc. It's probably not very practical and I'm mostly "reinventing the wheel" but my intention is to learn about digital synthesis in depth. So far I have successfully implemented basic concepts like oscillators and modulation of their properties. The next logical step is a filter.
So my question is: How does a digital filter work on this low level? How exactly does it modify the individual samples?
I understand, that this involves a lot of math. That's not a problem for me. I only need a good starting point (some sources to learn) and an intuitive explanation as all I was able to find were either analogue explanations or just formulas explained with a lot of advanced terminology that I'm not familiar with.