In the Public Sub in your class, you need a reference to the form concerned.
There are many ways of achieving this but one way is:
Dim _f As Form = My.Application.OpenForms("frmName")
An object of type Form does not, of course know anything about txtBoxName,
so you need to cast the reference to a variable of type frmName:
Dim _f As frmName = CType(My.Application.OpenForms("frmName"), frmName)
Now that you have the reference, you can manipulate the TextBox directly:
_f.txtBoxName.text = "The quick brown fox ..."
There are a number of nuances to this approach but if the app is a basic
Windows Forms app then this approach should work just fine in most cases.