What allows a composer to show what a charachter is thinking or feeling even though something else is sung?

The answer to this is Leitmotive. This is the process wherein composers would try to let the listeners know that the character in the song is feeling something different from what is being said. This is a theme that is still being done up to present time. How many times have you listened to a song and wondered if the song is actually being honest? This is the certain “it factor” that great composers have that cannot be faked.

They have the ability to convey the emotions that they want so that people will know and understand what the composer is truthfully trying to say. The singer may also play a role in letting you know what the song means but this is not related to leitmotiv.