This is all to do with friction (the title is a bit of a give-away really!). If you feel the placemat, you will notice that the picture side is the smoothest of the two sides, with the cork side feeling the roughest. Rough things have more friction than smooth ones so this means that the cork side of the placemat has lots more friction than the picture side.
When you pull on the placemat the smooth picture side glides over the beakers (because there's not much friction) whereas the cork side (where there is lots of friction) grips the tubes and carries them away with the placemat. So with the placemat and tubes gone, this just leaves the eggs dangling above the water-filled beakers. Gravity then does its thing and pulls the eggs down into the beakers, with the water providing cushioning for the eggs, meaning that they don't break...hopefully!
Remember that this experiment could get messy if the eggs break or the water spills, so do this somewhere safe.
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