Well $v_t$ here refers to the instantaneous speed of the subject particle and $a_t$ refers to instantaneous acceleration.

Let us look at this in 1D plane where x is the position vector of a small particle from the origin along the x axis.

We know very well that $$v = \frac{x}{t}$$where x is displacement in time t. ( Velocity is the rate of change of distance travelled in a particular direction)

Now , if we consider a small distance dx travelled in a small time dt by the particle ( along the x axis of course ), the velocity of the particle at a time t is ( Again, rate of change of displacement w.r.t time ) : $$v_t = \frac{d\vec{x}}{dt}$$ This is basically the instantaneous speed of the particle at a time t when x is a function of time t

Now since acceleration is the rate of change of velocity, the same applies to it as well.

This works for 2D and 3D as well by separately resolving the vectors and using this for each individual vector.

What Prof.Lewin has done is the same, except here he has written $\bar{r}$ ( which is called the position vector ) in terms of $\hat{x}$, $\hat{y}$ & $\hat{z}$ which are the position vectors along x axis, y axis and z axis respectively. When we differentiate this position vector we do it by individually differentiating in each direction as it is the sum of 3 functions basically.

If you still haven't understood the math, then you can probably find more about it under "Application of Derivatives", but in a math book, probably not in a physics book.