Decompose video frames into slices
Each frame of your movie is a 2D plane indexed by $x$ and $y$.
When you consider several consecutive frames, you can stack them in order to have a 3D cube, where each slice of the cube is a frame.
This space-time volume is now indexed by $x,y$ and $t$.

You can manipulate this volume as any 3D volume, just replace i nyour mind $t$ by $z$ if you find it confusing.

Once you have this volume, if you change the normal axis used to define the 2D slices (remember it was the $t$ axis initially), you obtain the different slices that you want:


choosing the $x$ axis as normal, you obtain 2D $(y,t)$ slices
choosing the $y$ axis as normal, you obtain 2D $(x,t)$ slices


By the way, the classical term for these kind of slices is epipolar-plane image analysis if you are loking for more references.