Here is an example of a pick-me statement.

"I am not like those other girls^1 that spend all day in front of a mirror^2. Instead, I am not afraid to get messy and will play sports with the best of them^3."

^1: This is a group that the speaker is a part of. It's typically one that is disadvantaged in some way. In this case it's women (or "girls"). Either way, the speaker will attempt to relate to the audience's group at the expense of their own.

^2: This is an action or other thing about the group. It is usually something that is harmless, like grooming oneself. In the above example, the speaker characterized the act of grooming oneself as something vain and not worth their time.

^3: This is something that the speaker believes the audience of this statement values. In this example, the speaker is attempting to appeal to the audience's group by saying they like sports and they don't care if it gets them dirty. This implies that people that groom themselves would not consider playing sports or doing something that would get them dirty. (Which is not true. Many women that care about their looks also play sports and get dirty.)

A pick-me will tear down their own people to fit in with another group. A woman being a pick-me with men has internalized misogyny, because she believes that she must separate herself from other women to have worth to men. (Instead of believing that women have worth as they are.)

This is a behavior you are likely familiar with, but never had a name for it. As with many things like this, sometimes the term is used on people and situations where they do not apply.

For instance, a woman who does not wear makeup and engages in many traditionally masculine hobbies, but does not tear down other women, may be falsely accused of being a pick-me.

One can apply the same logic to trans folks. I can't speak as much on how this would manifest, but conceptually it would be similar.