I've had this happen also. I think the process is sometimes an automated one - I have a video where I am playing my own arrangement
of a famous theme and it has a copyright claim on it! I don't claim to be an expert on copyright law, but I have had to do a little research as a consequence of giving public recitals, so here are a few general principles (from UK law - US law may vary but I imagine not substantially):
Firstly, I think there is a general assumption to view YouTube as an extension of a concert hall: a 'virtual concert hall' if you like. When playing in public, you should submit (to the promoter or appropriate body) a list of your pieces, the dates of the composer, the publisher of the edition you are playing frrom, and the edition date. This applies even if you are playing from memory
. Some places aren't as strict, and require you only to sign a declaration as to whether you are playing music which falls under the copyright remit or not. The concert is rendered liable to a fee, payable to the appropriate performing rights body, if any of the composer(s) performed has died within the last 70 years or if any of the editions nominated have been published within the last 70 years. (The 70 years specification is, I believe, applicable to the European Union, but it is 50 years in the US - though there have been moves afoot to elongate the duration). Please note that music is defined as a new edition even when there has been a minor editorial change made to a long-established text. Fees are even applicable for piped music in, e.g. dentists' waiting rooms. YouTube doesn't charge fees, but does entitle publishers to make claims. In your case, I cannot see how a publisher can "know" that they have used their edition when you are playing from memory, and the claim strikes me as borderline facetious. The MPRCS claim is probably some sort of generic claim, and I would suggest you check your videos to see whether they conform with the above general principles. I should add that YouTube's position is also odd to me in the UK, because there is an exemption from copyright fees if the concert is one where no financial transactions take place - i.e. free admission and (unbelievably this is crucial) no charges for refreshments at the interval.
Regarding appealing against the claims, I can't help you there. I hope they have not forced you to take the videos down - that has not happened to me on any of the videos I have had claims against. I've not bothered with an appeal, but it does strike me as exceptionally odd and haphazard when something like this happens, particularly when you bear in mind just how rampant flagrant copyright infringement is on YouTube.