Well, a monologue is someone speaking aloud, often to themselves. It’s almost like thinking out loud. So “Interior” monologue is the same thing, but it’s internal. It’s silent. It’s the thought process that we don’t let past our lips.
Why is this helpful for animators? Well, because if our hardest job (and it IS our hardest job) is to create a believable feeling that our characters have an internal thought process, then figuring out the interior monologue of a scene gives us actual thoughts to key off of, and actual changes in though process to base our acting decisions on.
Let’s say that in the scene, a man and a woman are arguing, and he’s jealous of the way she’s been flirting with a friend. Her line is “I love YOU,” with the emphasis on “you.” So we know that what she means is “I don’t love him, I love you.” When you are animating to that line, you could say that the monologue is “I love YOU,” and the *interior monologue* is “I don’t love HIM! How could even think that? Don’t you even know me?”
Now, when you are working out the acting decisions, you can treat the interior monologue just like actual lines of dialogue, and you can time your head shakes, blinks, searching eye movements, etc – you can time all that stuff off of this imaginary line that isn’t ever heard, but through your animation, we will FEEL it.
And *that* stuff is the meaty stuff that will bring your character to life. 
Meet Chloe – The secret life of pets.
She wants to eat all the food but knows she shouldn’t and is conflicted about it – after a little back and fro she just has to eat all the things!! Which is funny for the audience.
Creature comforts – Aardman
I really love this series, the voices are recorded and then the animations come later. Its stop-motion animation, the character designs are fantastic and the expressions and range they have is fantastic.
Inner Monologues – Robot Chicken
 – http://www.animationtipsandtricks.com/2008/10/what-is-interior-monologue.html