How can fallopian tubes become blocked?

Many ways: Fallopian tubes can be blocked for several reasons: a prior infection of the tubes, appendix, or uterus, previous surgery causing scar tissue inside the belly, and endometriosis. Sometimes surgery can provide answers by looking for these causes, and sometimes there is no obvious answer.

Also 'clogging': Most key reasons are covered above; many women get blockages near the opening from the uterus to the tube due to clogging with mucus, clot, endometrial debris, etc...These obstructions are mostly correctible by tubal cannulation viahysteroscopy, or selected cases by cannulation under fluoroscopy. The idea is not that different from a kitchen drain getting plugged during normal usage.