Copy&paste respects the current time indicator and is relative to it, which probably means your keyframes simply are somewhere far, far on the right side... It's always best to place the CTI at the start of the comp or layer, respectively, to retain correct timing.
I had thought the same thing, but I searched from beginning to end and there was nothing there. It's like it pasted the paint, but won't show any proof that it its actually there.
There are a couple of problems with his tutorial and there is a much easier workflow. First problem is the way he tried the in and out point of the shot and slid the layer to the first frame of the comp. I would have duplicated the layer, then after the painting was completed I would have simply created a new black solid in the main comp, deleted it, selected the layer I painted on and then selected the black solid in the Project Panel and held down the Alt/Option key while dragging the black solid to the Main comp to replace the footage. Works every time and solves the in shifted in and out point problem caused by the tutorials poor technique.
To see if the real problem is timing simply create a new solid, paint on a few strokes, then move back to the timeline move to the first frame, select the Paint effect in the Effects Control Panel and copy, add a new layer, select the layer and paste.