I do not understand how you can have acceleration without changing position (at 10:06). Acceleration is the derivative of speed, which is the derivative of position change. If the position change is zero, how can the acceleration be non-zero?
Position is not an absolute notion: you need to answer "position with respect to what?".
If the thing you're measuring position against is also accelerating, then you need to apply some acceleration of your own to stay still with respect to it.
The terms you want to look up are "proper acceleration" and "coordinate acceleration". The curvature of spacetime means the thing I'm measuring position against is moving relative to me (c.f. the example of two people walking in parallel across the Earth, nevertheless eventually meeting: the curvature means that even though neither of them is measuring an acceleration, nevertheless they are accelerating towards each other), so I need to have some internal ("proper") acceleration of my own to counteract the fact that our geodesics are moving away from each other.
Your position in spacetime is changing. You're going straight in spacetime, but spacetime is curved by the mass of the Earth so you're following that curve into the center.
The surface of the Earth keeps you from actually falling in, and is therefore pushing you away or upwards from the center. This is the acceleration acting on your straight line path through curved spacetime. This is the deviation from your geodesic.
