Alan Watts is where I learn about Indian philosophies and religions. The guy gave scores of lectures and all of them are easily understood and many are peppered with humor.
He held a class with young students and asked them, "What is a thing?"
And a bright girl raised her hand and said, "It's a noun."
And that's true: the root origin of "thing" is a "think" -- a speech convenience for identifying a cut-out from a unified world that we don't want to label.
So perhaps we should revise your teacher's phrase to:
"it's not that nothing exists, it's that no thing exists really"