Shakespeare's histories provide important social commentary
Although they contain many historical inaccuracies, Shakespeare's histories are very important for displaying social attitudes and conventions on issues such as politics and the monarchy, both from people in the medieval era, when the plays were set, and Shakespeare's own time.
Shakespeare's history plays present a huge range of characters from a variety of different social stations, from kings and princes to Falstaff, a poor drunkard who appears as a fictional character in many of Shakespeare's history plays. This variety allows the plays to make interesting observations about society and its structures, and how rich and poor people are both still human and fallible. Although many of those observations, especially those concerned with the monarchy and concepts of Englishness (especially important in Shakespeare's Henry plays, where Britain is fighting against France), are more relevant to Shakespeare's own time than the medieval era the plays are set in, they are still very interesting for scholars and audiences because they reflect political views of Shakespeare's era (especially those of the common people who usually could not read or write so left no legacy of their own). The history plays are uniquely able to do this because unlike the comedies or tragedies, they have a cast of characters from wildly different social positions. The comedies and tragedies tend to focus on the lives and fortunes of small groups of people, usually nobles, but the more sweeping national scale of the histories makes them well placed to comment on social issues and thus proves their historical importance regardless of how accurate the events they depict are.
The social commentary of the history plays is not necessarily reflective of actual society, but Shakespeare's perceptions of it and those of his sources. He relied heavily on Raphael Holinshed's chronicles of medieval history, which themselves were often embellished, and Shakespeare certainly emphasised the differences between social classes for dramatic effect, as well as entertainment given his large working-class audience. Additionally, the histories are not the only plays of his to contain social commentary - the working men of A Midsummer Night's Dream provide an important contrast to the noble families, for example - although admittedly the other genres do not use it as such a central theme.
The history plays present a huge range of characters from different class backgrounds, so they are able to comment widely on social issues and the views of society. Although these views are often more reflective of Shakespeare’s own time than the medieval era he was writing about, they are still interesting and important. The smaller casts of the comedies and tragedies mean they are not able to comment on social issues to this extent.
Rejecting the premises
Shakespeare's social commentaries are not necessarily accurate. Other genres of his plays also use class differences as plot points, not just the histories (although the histories do it most frequently).