See my answer to the conformal group of $S^n$ where you can find explanation for the the positive definite case and a link to a paper that actually works in arbitrary signature.

Here is a short summary of the argument: Rewrite the problem using definition of the pullback as a system of PDEs for $F$. Differentiate these equations and take clever sums. Use the original system to keep only derivatives which are not determined by the original equations. Repeat. Lo and behold! You obtained no more "undetermined" derivatives! It then follows that the solution space is finite-dimensional and you can write it's basis explicitly.

It's roughly a one page calculation plus half a page of text. For details, see the article by Slovák that I link to in my previous answer.