Can someone explain how the numbers on an anvil translate to the weight?
They don't always "translate" the same. It depends on the maker and the vintage.
On Peter Wrights and some other brands using the 3 number marking system, the first number is the English or Imperial long hundredweight (112 lbs per unit). Second number is quarters of long hundredweights (28 lbs per), and third number is pounds.
Sometimes a single number and fraction of long hundredweight is used instead. Example: 1-3/4, which would translate into 112 + 84 = 196 lbs. Such an anvil might also say "1-3/4 CWT")
Some American-made and newer anvils use the US Standard (and Canadian) short hundredweight (100 lbs per unit) instead of the Imperial long hundredweight.
For historical reference, the long and short hundredweight relate to other units in the Imperial and US Standard measuring systems as follows:
8 stones equal one Imperial hundredweight and 20 Imperial hundredweights equal one long ton (2240 lbs). The English/Imperial stone (14 lbs) is still a common weight unit in the British Commonwealth in spite of the Metric system.
20 US Standard or Canadian hundredweights equal one short ton (2000 lbs).