Capacitors, just like many other things in the world, can be understood in more than one way.
Imagine an un-charged capacitor. Imagine that you connect it to a DC power supply. What happens? The capacitor becomes charged. How? It becomes charged by a pulse of current that flows, just for a moment, pulling some electrons from one plate, adding some electrons to the other plate. No electrons flow through the dielectric that separates the plates.
Now imagine that you reverse the connections. What happens? You get another pulse of current that pulls electrons from the formerly negative plate, giving it a positive charge and, adds electrons to the formerly positive plate, giving it a negative charge.
Now, finally, imagine that you connect the capacitor across an AC power supply. The polarity of the supply is continually reversing. You get continual pulses of current. It appears as if the capacitor is "conducting AC current," when in fact, no current ever flows across the dielectric.
Electrical engineers who work with AC signals find it convenient to drop the "as if," and simply pretend that the capacitor "conducts AC current" and, "blocks DC current." It's a simplification of the underlying reality, and it is a simplification that is useful to them. It helps them to get their job done more quickly.
Literally everything we know about the universe is, on some level, a simplification. And, many things can be usefully understood on more than one level. One difference between physicists and engineers is that engineers, who are mainly interested in getting something done, prefer to look at the highest level that is not too simple to be of any use. It helps them to get the job done faster. Physicists, on the other hand, who are mainly interested in expanding our knowledge of the universe, tend to spend more time looking at the "deeper" levels.