One usage of Function is in Streams. Everyone uses map method these days, I believe:

This map method accepts the Function as a parameter. This allows writing a pretty elegant code - something that could not be achieved before Java 8:

Stream.of("a", "b", "c")
   .map(s -> s.toUpperCase())
   .collect(Collectors.toList());
// List of A, B, C

Now its true that there are method references and functional interfaces (one of which is Function of course), this lets you using method reference to rewrite the above example as:

Stream.of("a", "b", "c")
    .map(String::toUpperCase)
    .collect(Collectors.toList())

... but that's only a syntactic sugar - map still accepts the Function as a parameter of course.

Another example that uses Function from Java itself is StackWalker: Here is an example:

List<StackFrame> frames = StackWalker.getInstance().walk(s ->
    s.dropWhile(f -> f.getClassName().startsWith("com.foo."))
     .limit(10)
     .collect(Collectors.toList()));
}

Note the call to walk method - it accepts a function as a parameter.

So bottom line, it's just yet another tool that can help the programmer to express his/her intentions. Use it wisely wherever appropriate.