Be warned, there's a "woo-woo shit" risk factor here, which my skepticism keeps at arm's length. I'm more of a neuroscience / mindfulness meditation kind of guy. But I do cherry-pick from other areas, where my curiosity takes me. And the original quotes were good cherries.
Reframing the "nothing is ever personal" idea in more neuroscience terms: some astonishing high degree of our neurological processes (90+% ?) are subconscious or preconscious. A similar percentage of neurons are formed before the age of 18. In many ways, the quest to improve ourselves reduces down to the skill of paying slightly more attention to the activity of our minds.
So when someone interacts with you in a way that causes you stress or hostility, you can choose to recognize the above facts as playing out in the arena of their brain, in the same way as they are playing out in yours.
This is not to excuse behavior, nor disregard the need for boundaries, protection, standing up for yourself, etc. But it does take the sting off. What's better for your own equanimity? Succumbing to a feeling of being singled out? Or recognizing your counterpart as being stuck in their own behavior loop, unaware that they're (arguably) in a state of some kind of suffering?
Socializing is our most complex and wonderful skill; there are a ton of attendant instincts that evolved with it: status signaling, negotiation and exchange; hierarchies for coordinating group actions; grudges and revenge as deterrents meant to preserve social harmony (see chimpanzee behavior; then see bonobo behavior for something more inspiring). All of this monkey software can be dialed down, even outright idled at times. Because nothing is ever personal.
These are some truly advanced and empowering concepts, so apologies if I'm probably not representing them properly.