It’s an assumption that we make. If we see small fluctuations in the data, that might simply indicate noise. If we see large perturbations in the data, we believe that these larger perturbations have meaning.

If you take the “cloud” of data and plot it in p-dimensional space, then our first principal component will be in the direction of the line of best fit through the data. We abstract that to multiple dimensions so that the second principal component will be in the direction of the line of best fit in our cloud of data given that it’s orthogonal to all other principal components and so on. It wouldn’t make sense for our principal components to be aimed in these directions if we didn’t assume these larger variance directions were meaningful.