Consider the example of a hero being chased through a ravine by bad guys. Imagine if a boulder suddenly toppled from the cliffs and blocked the bad guys, allowing the hero to escape. It seems weak - a contrivance to get the hero out of a tough spot. However if the boulder fell in front of the hero, trapping him with the bad guys and making his situation worse, it feels like good drama.
This is because on some subconscious level the audience is aware that someone is crafting this story. But they don't want to see the hand of that storyteller at work. There are unwritten rules that the audience expects the storyteller to abide by. A coincidence that saves the hero feels like the storyteller wrote themselves into a situation and couldn't figure a way out, so they made something random happen to solve their problem. It feels like cheating. If the writer is allowed to let anything happen at any time then there can be no tension.
That type of coincidence is pretty easy to spot. More insidious is when a story element is a little too convenient. It exists to make the writer's life easier rather than growing organically out of the premise. But sometimes it's possible to turn these types of coincidences into strong plot points.
That's a little confusing in the abstract. Let me give an example of what I mean. One benefit of teaching is that it gives me ample inspiration for blog posts. Recently a student turned in an outline where when the hero meets the villain, they discover they are old high school classmates. I asked why that was and the student said, "I wanted them to know each other so the hero would trust the villain."
That's a valid idea, but it's imposing the relationship on the characters because of a plot need, rather than growing the plot out of the relationship between the characters. Thus it seems a little convenient.
I suggested that maybe the high school relationship between the characters could be the reason they come together. Maybe the villain was seeking out this old classmate, rather than running into them coincidentally.
My student's eyes lit up. She immediately began spinning a backstory of high school jealousy and rejection. Suddenly there was a whole new subtext to her main story. A weakness had just become a strength.
In good stories plot should grow out of the character's goals and relationships. But of course you need certain things to happen in order for your story to work. When you discover something happens in your story just because you, the writer, needs it to, try to think of a reason based in a character's goals for the plot point.
Let's go back to our falling boulder idea. The audience may accept a random boulder trapping the hero. It makes the scene more dramatic and exciting. But what if one of the villain's henchmen pushed the boulder down? Maybe they had been eavesdropping on the hero and knew his planned route. That feels more satisfying, doesn't it? Even when you could get away with a coincidence, motivating it with character is better.
The exception to this rule is when the entire story is based on some kind of random event. In these cases the randomness is part of the thematic underpinnings of the story. For example, what happens when a regular kid finds a bag of money? Or when two musicians witness a mob hit? Or when a businessman is mistaken for a criminal? The audience is willing to buy one "miracle" to set up the story. But every plot point after that should grow out of that random event and the characters.
In real life random things happen all the time. But in drama we want things to progress logically. Thus the saying, "Truth is stranger than fiction." So turn your coincidences into plot points based on character motivation. It might just enliven your whole story.