A forward curve has many points, and they move together. Principal component analysis extracts the independent patterns of movement, and for interest rate and commodity curves the first three almost always have the same interpretation: a level shift where the whole curve moves together, a slope change where the front and back move in opposite directions, and a curvature change where the middle moves against both ends.
The value of the decomposition is that it turns a many-dimensional object into three interpretable numbers, and those numbers correspond to trades. A level move is what an outright position captures. A slope move is what a calendar spread captures. A curvature move is what a butterfly captures. So the factor decomposition is not merely a summary; it maps onto the instruments a desk actually uses.
The relationship to Nelson-Siegel is worth stating because they are two routes to the same three shapes. Nelson-Siegel imposes level, slope and curvature as a functional form and fits the coefficients. PCA extracts whatever patterns the data actually contains, and then reports how much of the variance each one explains. The second is the honest one about whether the decomposition means anything, because it can tell you it did not work.
That explained variance share is the correct gate, and it is a specific instance of the general rule from z-scores: gate on the quantity that states the condition directly. If the first component explains 90 percent of the variance, 'level' is a real description of how this curve moves. If it explains 40 percent, the curve is not moving in a low-dimensional way and reading a level factor from it is imposing a structure the data does not have.
There is also a practical data requirement that the curve makes awkward. PCA needs a consistent set of variables through time, and a forward curve's contract months roll every month. So the decomposition has to be run on a fixed tenor grid, which means the curve has to be fitted first, and any error in that fit propagates into the factors.