Edit: I see that Ian has already given this answer. Nevertheless, I'll leave this here because it offers more detail. This is the closest thing to a real-world application of an existence proof of which I am aware.
Numerical PDE have many applications in modeling physical problems and help us accomplish many engineering feats - for example, building a bridge or placing an oil rig in the ocean.
To be confident in the answers that these numerical methods produce, we must know that the equations involved are well-posed, meaning
1) the solution exists,
2) the solution is unique, and
3) the solution depends continuously on the data given.
If (1) isn't true, then the model is flawed. If (2) isn't true, then which solution are you converging to? If (3) isn't true, then your discretization of the problem may introduce significant errors.
So for numerical PDE to work (and their applications in engineering), non-constructive proofs of existence, uniqueness, and continuity are used.