There are three parts to this answer: why slice, how to slice, and ‘what does it mean for time to seem continuous’. You can also review first (wikipedia) if it’s easier.
First, we may want to slice because a spectrogram (e.g., in Praat) is a 3D figure on a 2D display. Humans aren’t great at reading 3D figures. If you slice that 3D figure (intensity by frequency by time), you get a 2D figure. For a spectral slice, that’s (intensity by frequency); slicing lets us set aside time for the moment, and focus on the intensity-frequency distribution.
Second, this kind of slice is made by ‘cutting’ the spectrogram at a particular point in time (i.e., on the x-axis). The metaphor may be clearer if you think of the spectrogram as a real 3D object, like little plastic mountain range. If you use a knife to cut a perfectly vertical slice of that mountain range, you can see the silhouette by looking at the end. That outline is the spectral slice: a simple plot of intensity (y-axis) vs. frequency (x-axis).
Third, why even talk about time being continuous or not? The answer is that a spectrogram really is a big stack of slices put together. Mathematically, that’s where you get it (Fourier transform). In order to convert a single complex sound into a spectral representation, you’re forced to cut time into discontinuous pieces and analyze each little slice. The length of this slice in time is the window length (Praat manual).