Given a statistical model y = X₁b₁ + X₂b₂+ e
where
y is a vector of values of a dependent variable,
the X's are linearly independent matrices of predetermined variables, and
the e's are errors, we could premultiply the equation by M₁=I-X₁(X₁'X₁)^-1X' which projects vectors in the space spanned by X₁ to zero, and run OLS on the resulting equation M₁y = M₁X₂b₂+ M₁e
and (the theorem says) would get exactly the same estimate of b₂ that OLS on the first equation would have given.
This use of premultiplying is used in the derivation of many estimators: notably IV estimators and FE estimators.