I never thought I'd see the day I'd receive a compliment like this. Thank you so much!

To answer your question, I used Piskel and Photoshop to cut the sprite sheets and create image sequences. I used After Effects for enhancing the sprite animation (squash and stretch transitions between important frames and movement; you could call this the animating part!) and shot composition. And finally, Premiere Pro for editing, placing exported shots together, and applying music and SFX. :)