A Bezier curve is defined not by points it passes through, but by a cage of control points that pull it into shape. The two middle points are handles: the curve leans toward them but never touches them.
de Casteljau's algorithm builds the curve by repeated averaging. Slide the same fraction t along each edge of the cage, join the new points, and repeat — the nested lines collapse to a single point that rides the curve.
The direction of that last little segment is the curve's tangent, so the handles set not only the shape but the exact heading at each end.
Chain two curves and they meet smoothly only when the handle leaving the joint points exactly opposite the handle arriving — three points in a line. That trick is behind every font glyph, vector logo and animation path you have ever seen.
Something in the simulation stopped unexpectedly — the lesson continues without it. Nothing you did was wrong; you can move on.