Linear interpolation is used as a method used in mathematics of constructing a curve that has the best fit to a series of points of data using linear polynomials.

Interpolation in general is a way to determine intermediate values from a set of coordinates. Linear interpolation would be to fit a single linear function to the entire set of coordinates. Piecewise linear interpolation would then be to determine intermediate values from the set of coordinates by fitting linear functions between each set of coordinates. Connect-the-dots so to speak.

Advantages over what? For what?

Generally linear interpolation is done because one infers that the relationship between points is linear and/or it is the the easiest kind of interpolation. In the absence of data or theory to help you infer the relationship between points the principle of parsimony suggest that use the simplest that gets the job done - linear.

pu = p0 + u(p1 - p0)

The process is called interpolation, which applies a computed formula of the line to a given x or y value.

(More specifically, it is "linear interpolation".)