So I just started reading on FIR and IIR filters and am amazed by how "simple" the theory seems, so far.

* But what confuses me is, why does filtering work by creating a weighted sum of previous samples?
* What intuition makes one think this can produce desired filtering effects?
* It seems a bit unintuitive to me eventhough anyone can verify that summing delayed signals together does produce comb filtering. But desirable filtering? Why?