Here are some examples:
* "I couldn't get a word in edgewise with my aunt - she was telling me all about her trip to Hawaii."
* "The meeting was so chaotic, I couldn't get a word in edgewise."
* "He was so excited about his new job, I barely got a word in edgewise."
The image is that of trying to squeeze a word into a conversation that's already full of other words, like trying to wedge yourself into a crowded space.