These early proof sets were not normally issued to the general public but were given as gifts to government officials and dignitaries and some, like the 1890 sets, only contained the cent and the nickel. The list above is sets put together by collectors and represent all or most of the proof issues of each given year.
The above list is the Top Ten sold values for United States Proof Sets 1856-1914 and the number one proof set on this list is the most this proof set date has ever sold for publically.
These values do not apply to raw sets or proof sets in any other holder; these values only apply to proof sets graded and certified by PCGS (Professional Coin Grading Service) or NGC (Numismatic Guarantee Corporation). Also, a few of these sets holders have CAC (Certified Acceptance Corporation) stickers and this is considered further assurance that the set is the grade and condition stated on the holder's label.
You can't buy or sell United States 1856-1914 Proof Sets for the prices listed in this website unless they're in the same grading company holder, same grade, variety and condition, even then there's no guarantee your proof set will sell for the same value.
If you want to learn the value of your United States 1856-1914 Proof Sets, then you must know the proof set's grade, condition, and if it's authentic. You do this by taking your proof set to a reputable coin dealer or sending your proof set to PCGS or NGC (fees do apply with these services). If you're not interested in sending your proof set to a third party grading service then you should do some research and find a reputable coin dealer instead. I recommend a PCGS Coin Dealer since at least three of their piers have to vouch for them to be a PCGS Dealer.
If you can't find a PCGS Coin Dealer then look for local ANA members or coin clubs. In the mean time you can get an idea of your US 1856-1914 Proof Sets value and grade by visiting CoinHELP! website.