As has been said, the main point is to give up the idea of communicating your actual research topic. Even job seekers giving colloquium talks should usually not attempt this. The best such talks instead teach the audience, even mathematicians, the simplest underlying ideas of their subject, and only mention the direction of their own work in the last few minutes or so.
When discussing infinity with laypersons, I have often used "Hilbert's hotel", in which the infinitely many rooms are all full when another guest arrives. Everyone moves up one room and the new guest goes in room 1, thus showing that infinity plus 1 equals infinity, (as a cardinal). The audience easily figures out how to add 2 new guests or a thousand. Next ask them how to deal with an infinite sequence of new guests, which if course may be all placed in the odd numbered rooms, as the current guests each move from room n to room 2n.
When teaching topology I asked how to tell if an invisible butterfly net actually enclosed the butterfly, by looking at the winding behavior of the visible border of the net. when discussing higher dimensions, it is easy to get people to eventually visualize a 4 dimensional sphere as a family of three dimensional spherical slices, by starting with lower dimensional cases. Noting that one can escape a circle in the plane by jumping over it in three dimensions, point out that if one goes back in time, before the building one is in was built, one can escape a three dimensional room without breaking down the walls or opening the doors.
You will enjoy thinking about your favorite fundamental idea underlying your area.
edit: To convey the idea that knowing "how many" differs from knowing when two sets are "equipotent", I used the example of the cyclops in Ulysses, who knew when all his sheep were back in the cave, by matching them up one to one with a pile of rocks. Nonetheless he did not know "how many" sheep he had. I.e. "what number?" differs from "same number".
