I'm referring specifically to the $\frac{\delta f}{\delta x}|_{y}, \frac{\delta f}{\delta y}|_{x}, \frac{\delta u}{\delta x}|_{t} and \frac{\delta u}{\delta t}|_{x}$. I've included the rest to provide some context.
Edit: They also provide an example which confused me. Could someone also explain how they did the example?:
This is sometimes used to mean "keeping this variable constant". It emphasizes that while you take the partial derivative with respect to e.g. $x$, you are keeping $y$ constant.
I think it has roots in certain areas of applied math where there may be constraints. For example if you have pressure, volume and temperature of a gas, then they always satisfy a constraint. So when you write the partial derivative with respect to say, pressure, do you let the volume and temperature change with the constraint? (physically natural but not strictly speaking what the usual partial derivative on $\mathbf{R}^3$ is) or just keep them constant? (violates the gas equation but is mathematically natural). Here the notation would be telling you explicitly to do the latter.
I don't like this notation because often it is completely redundant, but sometimes it is absolutely crucial.