1. What is the joint angle velocity?

The joint angle velocity is the angular velocity of you joint, because we are acutally dealing with a 2D problem. This will become clear if you look at your system from the $z_0,z_1$. In the picture this angular velocity is given by $\dot{\beta}$.

2. What is this time-variant rotation matrix?

You have to consider four coordinate frames. The first coordinate frame is fixed at your joint axis $K_0=(x_0,y_0,z_0)$. The second frame is frame 1 but now rotating around the $z_0=z_1$ axis. The other two frames are there in your inital image. If you tranform from frame 1 to the other frames you will see that the rotation matrix will be time dependent. This is obvious, because depending on the time dependent motion of $\beta$ the coordinate frames will also move with time dependence.

3. Why do the angular velocities differ only by the joint angle velocity and a time-variant rotation matrix? What does it even mean, actually?

This means that the angular velocity of the left coordinate frame $\omega_1$ and the angular velocity of the right coordinate frame $\omega_2$ are related by the angular velocity of the joint $\dot{\beta}$. The relationship is given by:

$$\omega_2-\omega_1=\dot{\beta}.$$

Note, that this simple equation is only true because we only have a 2D rotation.Imagine $\dot{\beta}=0$, then the joint is acting like a fixed link, hence the left and right coordinate frame have the same angular velocity. If $\dot{\beta}$ is positive then the angular velocity of the right frame $\omega_2$ will be the sum of the angular velocity of the left frame $\omega_1$ and the angular velocity of the joint $\dot{\beta}$.

1. What is the joint angle velocity?

The joint angle velocity is the angular velocity of you joint, because we are acutally dealing with a 2D problem. This will become clear if you look at your system from the $z_0,z_1$ axis direction. In the picture this angular velocity is given by $\dot{\beta}$.

2. What is this time-variant rotation matrix?

You have to consider four coordinate frames. The first coordinate frame is fixed at your joint axis $K_0=(x_0,y_0,z_0)$. The second frame is frame 1 but now rotating around the $z_0=z_1$ axis. The other two frames are there in your inital image. If you tranform from frame 1 to the other frames you will see that the rotation matrix will be time dependent. This is obvious, because depending on the time dependent motion of $\beta$ the coordinate frames will also move with time dependence.

3. Why do the angular velocities differ only by the joint angle velocity and a time-variant rotation matrix? What does it even mean, actually?

This means that the angular velocity of the left coordinate frame $\omega_1$ and the angular velocity of the right coordinate frame $\omega_2$ are related by the angular velocity of the joint $\dot{\beta}$. The relationship is given by:

$$\omega_2-\omega_1=\dot{\beta}.$$

Note, that this simple equation is only true because we only have a 2D rotation. Imagine $\dot{\beta}=0$, then the joint is acting like a fixed link, hence the left and right coordinate frame have the same angular velocity. If $\dot{\beta}$ is positive then the angular velocity of the right frame $\omega_2$ will be the sum of the angular velocity of the left frame $\omega_1$ and the angular velocity of the joint $\dot{\beta}$.