Woman convicted of killing her mother prays for her innocence to be proved. It took some really fancy forensic work to overturn the bad forensics that had convicted her, but it was overturned in less than a year. The fancy forensics involved the discovery that her car make was known to catch fire from a faulty ignition design, and that melted plastic acts as an accelerant to fire.
There is no way to prove this is answered prayer. I think the example I gave in the OP where the answer met some particular conditions asked in the prayer was proved (she encountered the murderer and was instrumental in solving the case), but most prayers are like this one that is too easily dismissed as coincidence. But this is the most common kind of answer to prayer and if you are someone who prays you know it is God's answer. Proving it to someone else may be impossible.
Just as a side note it's interesting that this series "Forensic Files" includes such examples of people who were wrongfully convicted of a crime through bad forensic science. They protest their innocence of course which often only increases the suspicion against them. Getting anyone to take them seriously is the first big hurdle and then all the tests have to be done again and new kinds of tests added. So I could argue that this case I describe above shows answered prayer just because of the speed with which she got what she prayed for.
The context of that quote is how we get saved -- i.e. become righteous, are justified -- this became the pivotal reference for Luther who had been struggling to earn his salvation by good works and then learned through this Biblical teaching that God had done it for him and he had only to receive it by faith.