Does time dilation cause gravity as explained in this video?

There is much to dislike about the popularized video, but its basic idea isn’t far from the truth. If you work through the equations of weak-field GR, you will find that ${{g}_{00}}$, which measures the rate of passage of proper time (i.e., aging) along a world-line relative to coordinate time, can be identified with $1+2\Phi $, where $\Phi $ is Newton’s gravitational potential. The geodesic equation predicts the same deflection of trajectories as Newton’s equation of gravitational force.

How would a change in the passage of time cause deflection? It’s hard to understand for a point particle, and that’s why the video showed a dumbbell. There is an analogous problem in QM about the deflection of a wave packet moving through a potential gradient. It will have wave-fronts of constant phase orthogonal to the direction of motion. The potential affects the rate of phase change, so if the left side of the packet undergoes more rapid phase change than the right side, the trajectory will veer left.

The video just poses that time is curved around Earth, without an explanation.

Richard Feynman has given a nice explanation of why a mass falls down in a gravity field in which time is curved. The space part of spacetime, which only affects fast-moving objects (wrt the gravity producing mass), doesn't have to be taken into consideration.

Imagine two freely floating rockets (side by side) of height $L$ in outer space. The rockets have a light on top that gives a flash of light every second. A person who is attached to the floor of both rockets. They see the subsequent flashes one second after another too but a bit ($\frac{L}{c}$) later than the light that emits them. So the time at the top flows with the same rate as the time on the floor, according to both persons. Now one rocket starts to accelerate, parallel to the other, and at the same time, the lights on the top give a flash of light. The person on the floor of the accelerating rocket receives this flash of light a bit earlier than the person in the freely floating rocket. The rocket keeps accelerating and the lights give off the second flash of light. The person in the accelerating rocket receives this second flash not the same amount earlier as the first flash (otherwise the second person would receive the flashes one second after each other too), but a higher amount earlier as the first flash because the rocket is accelerating towards the second flash.
This means the person in the accelerating rocket sees that there is less than a second passed between receiving the two flashes. The same holds for all subsequent flashes. So for the person in the accelerated rocket time goes slower.
Now, according to the equivalence principle, there is no difference (locally) between a person sitting in an accelerated rocket (usually accelerated elevators are used to illustrate the principle, at the core of general relativity) and a person sitting on a mass (say, Earth) which gives objects the same acceleration as the acceleration of the rocket ($9,8\frac{m}{{sec}^2}$).
So time slows down in a gravitational field while objects fall down. The higher above the surface of the Earth you are, the slower the rocket has to accelerate to obtain the same acceleration caused by gravity, so time moves faster the higher you are (time is curved, which is also the case in a field with constant acceleration for every object, i.e. in a constant gravity field, though in this case of a constant gravitational field there are some difficulties see this article; it's quite complicated). Both the slowing of time and acceleration occur simultaneously and don't cause each other: it's just the Natural state of gravity. The slower pace of time, when going up in the field, and acceleration go hand in hand (in a parallel way, so to speak).
So (for non-relativistic speeds of a mass) it's not the curvature of time that causes an object to accelerate in the gravitational field of the Earth.

If instead, we would send a photon perpendicular to the accelerating rocket, a person in the rocket sees the photon bend down. Time is not involved because photons do not age. In this case, it's only the curvature of the space part of spacetime (if time is curved space is also curved and vice-versa) that goes along with a deflected photon in the gravitational field of a big mass.