Digital filter implementation
Digital filters perform mathematical operations on numbers. They're as "simple" as that. The complexity comes in the operations that are performed.

The simplest you can imagine is a basic "2x averaging filter" or 50% low-pass filter.  This takes a set of samples and halves the bandwidth, resulting in a sample set of 50% of the size.  Basically it involves taking successive pairs of samples and averaging them, resulting in a new sample that takes the place of the original 2, giving a 50% reduction in sample quantity, and thus a 50% reduction in sampling frequency.  That's the most basic low-pass filter.

So say you have a sample set of 1024 samples taken at a sample frequency of 16KHz (so a Nyquist-Shannon upper frequency of 8KHz), and you perform that simple low-pass filter, you then end up with a sample set of 512 samples taken at 8KHz, so a Nyquist-Shannon upper frequency of 4KHz.  Simple low-pass filtering.

Now, that how the digital filters work (in a very very small nutshell).  What you then do with those filtered sample values is up to you.  You may want to then return them back to an analog form, in which case they would then be passed on to a suitable DAC, or you may want to do more processing, or displaying of data, or other things.

That part is really down to the application.