This proverb uses a vivid image to illustrate the foolishness and danger of getting involved in an argument that doesn't concern you. Here's a breakdown:

"Like one who grabs a dog's ears...": This is the analogy. Grabbing a dog's ears is a bad idea because:
It's likely to provoke the dog.
It's painful for the dog.
You're putting yourself at risk of being bitten.
It's ultimately unproductive.

"...is one who passes by and meddles in a quarrel not his own.": This is the application of the analogy. When you interfere in someone else's fight or disagreement, you're like the person grabbing the dog's ears.
You're likely to escalate the conflict.
You're likely to get hurt (emotionally, physically, or socially).
You're likely to make things worse, not better.
You're not really helping anyone.

In essence, the proverb means:

Mind your own business. Getting involved in other people's arguments is a foolish and risky act that is likely to cause you harm and not resolve the situation. It's best to stay out of conflicts that don't involve you directly.

Key takeaways:

Danger: Just as grabbing a dog's ears can lead to a painful bite, interfering in a quarrel can lead to negative consequences for the meddler.
Foolishness: It is unwise and lacks discretion to insert oneself into a situation that is not their responsibility.
Provocation: Meddling can often stir up more conflict and make the situation worse.
Cautionary Tale: This proverb serves as a warning against unnecessary involvement in the affairs of others.

The proverb isn't saying that all intervention is wrong. Sometimes it's necessary to intervene in situations of abuse or injustice. However, it's warning against impulsive and uninvited involvement in arguments that are not your responsibility.