Grab the METAR data for MAN and FRA for the time zones your flight was meant to operate.

Use a METAR decoder to tell you what it means. Even a layman can get a pretty good impression, for example "Scattered clouds, light showers, 20 knot winds" is indicative of nothing unusual for the UK. If you go on AVgeeks forums you can get some more enlightened opinions.

Again it's not really on the customer to have to prove this, but if you approach the airline with this knowledge in hand, they will know that you know, and most likely will pay up.

And complain to the German equivalent of the CAA either way, because it's unacceptable the BS excuses that airlines are pulling to try and get away with their legal duties.