According to “The Theatrical World” (Braunmuller ix–xxi), which specific segments of Elizabethan society attended Shakespeare’s plays, and how could Shakespeare have written As You Like It to appeal to various people in his audience?

Shakespeare's plays were attended by several different segments of society depending on the venue. We cannot be entirely sure about the audiences of As You Like It in Shakespeare's day, as there is no definite record of a performance until 1599. However, we can make assumptions based on similar plays and performances. David Bevington gives the following description of the audience at an outdoor public theater such as Shakespeare's Globe:

For the most part, the audience was affluent, consisting chiefly of the gentry and of London's substantial mercantile citizenry who paid two to three pence or more for gallery seats . . . but the ample pit or yard also provided room for small shopkeepers and artisans who stood for a penny. The spectators were lively, demanding, and intelligent. . . . Shakespeare appealed to the keenest understanding of his whole audience. (Introduction to Complete Works of Shakespeare, xlv)

Later in his career, Shakespeare's plays were performed indoors to a more exclusively affluent audience. However, it is likely that As You Like It was performed in front of diverse audiences in Shakespeare's day.
The broad appeal of As You Like It can be attributed to the great variety it contains; Shakespeare incorporates a great diversity of characters, settings, and even genres into the play. In terms of social class, Shakespeare's comedies present the widest range of characters in his entire canon—and As You Like It is at the top of the list in this regard. Nearly all layers of society are represented in the play, ranging from the two Dukes and their families to peasants such as Audrey and Corin. The high-class as well as the lower-class characters are presented in both positive and negative lights. Duke Senior embodies the nobility and grace of the most benevolent rulers, while Duke Frederick shows the corruption and vindictiveness that can make the ruling classes cruel and unjust; likewise, Corin demonstrates the hospitality and kindness of rural folk, while Audrey is portrayed as somewhat foolish and naive. A theatergoer does not necessarily need to see members of his or her own class represented to be entertained by a play, but the broad range of character types in As You Like It keeps the action interesting for a wide variety of audiences.
It is not quite accurate, however, to think of the play's characters as separated into different and distinct social groups. The characters mix together, exchange roles, assume disguises, and transgress rules that would normally apply to people of their status. Duke Senior is forced to take up the life of a forester, Rosalind and Celia disguise themselves as middle-class travelers, and Touchstone happily exchanges court life for country life, among many other examples. Normal relationships between social groups are turned on their head for comedic, tragic, or romantic effect.
This disruption is a key trait of the pastoral mode, one of the literary traditions to which As You Like It belongs. Exceptionally popular in Shakespeare's day, pastoral literature usually features upper-class characters living by choice among rural people and becoming wiser from the experience. Alternately, pastoral literature may depict rural people as the equal of the urban elite in passion, humanity, and artistry. Often these works are intended to critique social norms or expectations by showing them in a new light. The disruption of normal social relations, and the opportunity for experimentation, play, and freedom is a hallmark of As You Like It and other pastoral works. The transformation of dukes into peasants, shepherds into poets, and women into men made the play compelling to all sorts of playgoers in Shakespeare's day—and continues to do so today.
Other relevant examples of the pastoral mode featuring the interaction of many layers of society include Shakespeare's main source for the play, Rosalynde by Thomas Lodge (1590); Arcadia by Sir Philip Sidney (1590); and Shakespeare's later play The Winter's Tale.