Well, not so funny since it's supposed to be a real java app to test your skills:

Can't the king also reach the other squares (eventually)?

I don't see what's supposed to be funny here. Obviously it's meant for a beginner who is just learning how the pieces move. But I don't see why it's funny.

Well, the oops, that's not right bit is slightly amusing

Perhaps that's referring to the application itself?

The king can't move anywhere!  It is an obvious checkmate that all must learn eventually, the feared Invisible Queen Mate!

I don't get it, what is wrong with the application itself? It looks like it's working fine to me. If the user had only selected squares the king can move to it would probably have told him that he's right.

The user has only selected squares the king can move to ... unless you count the box around e3.

Now I'm getting worried. Why cant the king move to the selected squares?

Yeah, I'm seeing the same thing as you Loomis -- I'm sure if the scenario had been answered properly it would not have given the error message.

Look closely. 

The king cannot move to e3. So the answer given by the user is incorrect. Oops!

Ah

Why wouldn't I count the box around e3? I count all the other squares with boxes around them. Honestly, I don't have much info to go on about how the application works.

To be honest, I was being purposely obtuse in my answers because the OP was so obtuse in his post.

I don't get what is so funny, the user clicked the square e3, which the K can't move to because he is on, and it told him he was wrong... What's to get? 

I got promoted to Baron and then to Count. I got annoyed by the sound but I do feel like a Grandmaster.