For example Satipatthana - or "mindfulness" looks at phenomena starting from sensory experiences in order to prevent cravings or aversion.
Cravings and aversion come from how you react to things you may find desirable or undesirable.
The desirableness and undesirableness are connected to you.
When you are able to reduce cravings or aversion in whatever way, you also affect how you relate to these events, and may also aid in not taking things personally.
Vipassana, another practice, guides you to see the impermanence of all things, as well as how your sensations and emotions are coming from natural phenomena rather than from this object you may call "yourself". By understanding the origin of how these relate a bit deeper, you can see better how there is no need to label a thing as "I" necessarily to be involved in any of this.
To credit my understanding a bit; I have studied and practiced western mindfulness, Satipatthana and Vipassana for years, as well as Buddhism in general.