I need to figure out how to find the number of times the word the appears in a sentence

for(int i=1; i<=str.length()-3; i++)

Because you start your for loop at index 1, you start analyzing the string at the second character, not the first. Java indices are zero-based.
I think the method indexOf[^] is far more appropriate for your requirement.

I'd recommend to use Regular Expressions. See: Java Regular Expressions[^]

import java.util.regex.Matcher;
import java.util.regex.Pattern;

public class RegexMatches {

   private static final String REGEX = "\\bcat\\b";
   private static final String INPUT = "cat cat cat cattie cat";

   public static void main( String args[] ) {
      Pattern p = Pattern.compile(REGEX);
      Matcher m = p.matcher(INPUT);   // get a matcher object
      int count = 0;

      while(m.find()) 
         count++;
      
      System.out.println("Number of matches: "+count);

   }
}

This code fails if word 'the' is first or last word in sentence.

Quote:

I found that the code down below works but it doesn't find the the at the beginning of the sentence

Your code do not behave the way you expect, or you don't understand why !

There is an almost universal solution: Run your code on debugger step by step, inspect variables.
The debugger is here to show you what your code is doing and your task is to compare with what it should do.
There is no magic in the debugger, it don't know what your code is supposed to do, it don't find bugs, it just help you to by showing you what is going on. When the code don't do what is expected, you are close to a bug.
To see what your code is doing: Just set a breakpoint and see your code performing, the debugger allow you to execute lines 1 by 1 and to inspect variables as it execute.

Debugger - Wikipedia, the free encyclopedia[^]

Mastering Debugging in Visual Studio 2010 - A Beginner's Guide[^]
Basic Debugging with Visual Studio 2010 - YouTube[^]

http://docs.oracle.com/javase/7/docs/technotes/tools/windows/jdb.html[^]
https://www.jetbrains.com/idea/help/debugging-your-first-java-application.html[^]

The debugger is here to only show you what your code is doing and your task is to compare with what it should do.

For use with debugger, change the code to

for(int i=1; i<=str.length()-3; i++)
{
   String c1 = str.substring(i,i+3);
   String c2 = str.charAt(i-1);
   String c3 = str.charAt(i+3);
   if(str.substring(i,i+3).equals("the") && !Character.isLetter(str.charAt(i-1)) && !Character.isLetter(str.charAt(i+3)))
         count++;
}

to see what are the values used in if while you debug.