I think he's saying that Legendre Polynomials form a sort of "function" space that you can make any function out of the Legendre Polynomials with that sum you wrote.
The integral you wrote is just a method of how to find out "how much of f(x) is in Pl(x)," so you can construct f(x) purely as a sum of legendre polynomials.
Different people have their ways of understanding this, but here is one I like to think of:
Think of the legendre functions sort of like vectors in some way. When you do that integral, you can think of it as a dot product (the limits never change and are important). If you think of it this way, this means that you are making f(x) a linear combination of "legendre polynomial" vectors. The neat thing about orthogonal functions is that their "dot product" (or integral : \int_{-1}^{+1} P_l(x) P_m(x) dx = \frac{2}{2m+1} \delta_{l,m}) is zero unless if it is with itself.
This way you can make an orthogonal function space where you can do some neat mathematical tricks. I have not done much electromagnetism, but I know a good example is in quantum mechanics where you need the eigenvalues of an operator. I don't know if you've seen this before but when an operator acts on its eigenvector, it gives an eigenvalue. So if you have an operator acting on an arbitrary vector, you need to break it into a sum of its eigenvectors (which by nature are orthogonal) to get the answer.
Typically, in quantum mechanics, the operators act on functions, so these "eigenvectors" are then functions, but usually sines and cosines (of the form cos(npix/L) for example, for a space from 0 to L), not legendre polynomials.
I don't want to go too much into this if you haven't seen this, but I do believe most likely what is happening is that it must be easier to tell something about the system if you break it up into its legendre polynomial "components."
Another example is with sines and cosines. By the Maxwell's equations, you know that electric signals are in the form of sines and cosines. If you receive a signal that looks like some arbitrary f(x), you can make an orthogonal space of sines and cosines and find out "how much f(x)" is in each sine or cosine component.
For example consider:
\sqrt{\frac{L}{2}}sin(\frac{n \pi x}{L})
on the interval 0 to L. Each value of n is a different sine function. This integral with find the sum that follows:
C_n = \int_0^L F(x) \sqrt{\frac{L}{2}} sin(\frac{n \i x}{L}) dx
for
\Sigma_{n = 0}^{\infty} C_n \sqrt{\frac{L}{2}} sin(\frac{n \i x}{L})
I hope this helps
