I do think it pays to have an understanding of Shakespeare's life and family and his plays and sonnets to get the best out of it.
For example Shakespeare's father held quite a few official posts in Stratford but he also got in trouble with the law a few times and got fined so he was rather discredited in later life as is mentioned in the show quite a few times.
There was also a "joke" in one episode about signatures of famous being valuable in later life and Shakepeare made a joke about never signing his name the same each time.
This is in fact true. We only have 14 words in Shakespeare's handwriting, and 12 of those are his first and last name, but in each case he spelt them differently.
There are also "jokes" about Shakespeare's lack of education (which is why some people don't think he wrote the plays). In a number of his plays he sets cities in the wrong place (an inland city on the coast and so on).
Also in his sonnets (which some people believe were private and never meant to be published) about two thirds of them are a man talking of love for another man, which is why in the shows he often makes "jokes" about not fancying the person who he wrote the sonnets for.
btw The title comes from Robert Greene (played by Mark Heap in the show) who also wrote plays, but who in his life published a pamphlet attacking Shakespeare and calling him "Upstart Crow"
https://en.wikipedia.org/wiki/Robert_Greene_(dramatist)
Anyone who fancies reading about Shakespeare's life then Bill Bryson wrote a fine fairly small book about his life. It also covered the history those times (Catholic and Protestant arguments etc) and the problems people had just surviving what with the plague and other illnesses.
Like all Bill Bryson books it is very well written and a joy to read with quite a few amusing lines and "jokes" in there. He really is a great writer.