Okay, when I first read your question, I was stunned that it was not obvious. Then the more I thought about it the more I saw how confusing this could be.
Imagine a small, fuzzy, wiggling puppy. All it wants to do is lick your face and be your friend. And it wants to do it RIGHT NOW, over, and over, and over again because it will never express enough how much it loves you.
Now apply that same single-minded, NOW-ness to expressing that you're really, honestly, completely sincere.
That's what's meant by "puppy-dog earnestness".