Firstly, I don't like the animation as it uses a square lattice. The Fourier transform of a square lattice is just a square lattice of spacing 2 * pi/a where a is the spacing in real space. In general, the form of a reciprocal lattice is different than its corresponding real space lattice and that video chooses the only example that really doesn't emphasize that.

To be honest I don't really know what you're trying to ask with your question. However, if the real space lattice is a "recipe" for how to reconstruct a regular periodic structure by giving directions to where each atom should go (position); than the reciprocal lattice is, in a hand-wavy way, a "recipe" for how to reconstruct a regular lattice using only sine-waves pointing in different directions, having different amplitudes and different wavelengths.

If I imagine a line of regular spaced points, my position-based "recipe" requires an infinite number of entries to describe that structure because there's an infinite number of points. However, if the lattice spacing is a than I only need ONE sine wave to describe the whole shebang and its wavelength is 2 * pi/a. That's a structure there's only one particle, A and thus I have A - A - A - A ... and each dash (-) is a distance of a. In one dimension all we have to keep track of is the number of sinewaves, the magnitude of the sinewaves and their wavelengths

Now for a higher dimensional example (2D or 3D) it is more complicated because depending on the angle you're looking at the crystal you see the lattice differently, which is to say if you want to make a 1D recipe of sinewaves like before, that recipe changes depending on the angle of viewing. Furthermore, one can't just pick a single angle and create a recipe and say you've completely described the crystal, as different crystal shapes can look indentical along certain "sweet" angles. Thus, to fully describe the 2D or 3D structure you need to include many angles. But the issue is "how many angle do I have to build 1D recipes for?"

The whole machinery of Bravais lattices and such is a mathematically condensed way of doing this in a procedural way using as few recipes as possible.