The raw curve is a set of discrete points: one settlement per listed month. Many things a desk wants to do need a continuous function instead. Interpolating a tenor that is not listed, comparing today's curve shape to a curve from a month ago when the months have all rolled, or separating the shape from the noise all require a fitted curve rather than a scatter of points.
The standard family for this is Nelson-Siegel, borrowed from fixed income. It expresses the curve as a level, a slope and a curvature term with a single decay parameter, which is a strong constraint: four numbers have to describe the whole curve. That is the point. A fit flexible enough to pass through every point would reproduce the noise along with the shape, and there would be nothing left to compare.
The fit is useful precisely because it is restrictive. Once a curve is four numbers, two curves from different dates are comparable even though every contract month has changed underneath them. That is what makes the parameters trackable through time, and it is the same idea that curve factors arrive at from the other direction: Nelson-Siegel imposes level, slope and curvature by assumption, while PCA extracts them from the data and then measures how much they explain.
There is a specific trap when contracts are filtered. If a curve build drops months that fail validation and then renumbers the remaining tenors 1 to N, every spread computed from the renumbered series is between the wrong pair of months, and the numbers will look plausible. Tenor identity has to survive filtering.