It’s a play on a more common expression, “to trip on one’s tongue,” meaning to have trouble talking due to nervousness. In your example, the metaphor is exaggerated even more. You’ve parsed the sentence wrong — it’s not “she didn't trip over your tongue dragging.” Here’s how it should be parsed: You're lucky / (that) she didn't trip over / your tongue (which was or as it was) dragging halfway across the ground. The word “dragging” refers to the location and movement of the tongue. You can look it up in the dictionary of your choice, although it really doesn’t mean much in this particular sentence. BTW, we usually don’t use the word “aphasic” in conversation. I’d suggest “tongue-tied” or “speechless” in this context.