It works because a new object is returned. The variable 'x' does not change.
public String replace(char oldChar, char newChar)
Returns a new string resulting from replacing all occurrences of oldChar in this string with newChar.

Warning: a mutable object declared as final will still be mutable. The only thing that final locks is the binding between the variable name and the reference of the object in memory, so final "protects the code" only when used on immutable data structures or basic data types:

final List<String> lst = new ArrayList<>();
lst.add("stuff");                              // => works like a charm!

If you try to assign new object or value to the variable with final keyword, the code will not compile.
You should use it, if you specifically do not want any other programmist (or yourself) to change the value, because some of your code depends on it. In example:

final double PI = 3.1415;
// PI = 10.0; this line would not compile

In this kata:
x = "my custom solution"; // would not compile