There are three broad questions here, together covering much of the field of structural bioinformatics. I will answer each briefly but point you to a textbook for more.
Why is predicting protein structure useful?
This is actually a very good question. The standard answer here is "drug discovery", but as things stand anything other than a high quality homology model isn't particularly useful for drug discovery. I can't think of any examples where de novo structure prediction has directly led to the discovery of a drug, for example by virtual docking in a binding site, though I am willing to be proved wrong. In the future though, as protein structure prediction and virtual screening both improve, this could end up being an important application of structure prediction.
Other current uses that are more developed are: A) protein design, where improvements in structure prediction allow you to find better sequences that form certain structures and carry out certain functions (the inverse folding problem); B) exploring the evolutionary relationships and function of a protein, e.g. if a predicted structure looks like all the other membrane transporters then it probably is one too (see more on that below); and C) running molecular dynamics simulation on the structure to gain biological insights and complement experiments.
On a deeper level, scientists will always seek to answer the question of what structure proteins fold to and how they fold, because it's just such an interesting problem that is central to molecular biology. Solving it will almost certainly lead to useful breakthroughs, even if their exact nature is unclear now.
How do we predict protein structures?
By the original formulation of the problem, protein structure prediction is arguably solved. If you can find a template, i.e. a related protein sequence with an available experimental structure, then you can pretty reliably get a high quality model (less than ~3 Å RMSD). Improving a model beyond this is currently called "refinement" and this will become increasingly important as we look to get ~1 Å RMSD models that can be used in place of experimental data.
If you can't find a template you can still have a decent go at the structure, provided you can find enough related sequences. It turns out that positions in a multiple sequence alignment will covary if the residues are close in space in the structure. Initially statistical techniques were used to extract direct from indirect coupling effects, but now deep residual neural networks are showing state of the art results in this field. These developments are recent and have been the focus of news reports. The explosion of sequence data facilitates this approach, though it is still not "the solution" to those who only want to use a single sequence as input data. For pure physics-based approaches there has been limited success on small proteins, see for example here, but these methods are not in broad use for structure prediction.
Usually the input to these methods is just the protein sequence, though you often bring in other data (templates, related sequences) as part of the pipeline. We generally care about the structure at physiological conditions, which usually corresponds to the structure found in x-ray crystallography or NMR, so predictions under different conditions are not yet routine. For more on protein structure prediction, see the CASP website and have a read of their papers.
How useful is protein structure at predicting function?
Predicted structure can be used to transfer function from related structures with known function - see for example here and here.
It is not currently possible to use a predicted structure to predict the function using chemical arguments, for example by saying "I have predicted a binding site with a certain arrangement of amino acids so this must have function X". However as structure prediction improves and we have more structures and functional annotations, this is an exciting prospect.
With respect to protein-protein affinities, if you have the structure you can start to predict and rationalise the structure of protein complexes. Such predictions from structure alone (i.e. not using homology to known complexes) are not routine yet, though more data and better models will improve this. See for example CAPRI. This is clearly a biologically important area, as most proteins form complexes.
Sequence determines structure determines function (fingers crossed as I'm simplifying quite a bit).
You shouldn't have to know the structure to predict function/binding from sequence, but it helps, and a sufficiently advanced system would learn this anyway in order to make the connection.
Protein structure prediction is a hot research topic that currently has limited applications, but is certain to have more in the future. If anything, it has only got more interesting over the last 50 years.