public boolean beginsHowIgnoreCase(String str)

Given a String, return true if it starts with "How". false if it doesn't.
The method is case-INSENSITIVE, that is, capitals (upper case letters) and lower case letters match.

Explanation:
This can be done with just two easy changes:

public boolean beginsHowIgnoreCase(String str) {
  boolean begins = false;
  str = str.toLowerCase(); 
  if (str.startsWith("how",0)) {
    begins = true;
  }
  return begins;
}

(1) Use toLowerCase(), which creates a version of the string where all letters are lower case.
(2) Change the 1st parameter in startsWith() from "How" to "how" (all lowercase).

Note that you need to RE-ASSIGN the value of this new string to the str variable:

str = str.toLowerCase();