she used to take it personally?

the last sentence above is your clue.
The childs used to (past tense) take personal responsibility for her father leaving. As a grown up she does not feel it is or was her that caused her father to leave.

Asha blamed herself for her father’s absences. She thought he left because of something she, personally, had done to upset him. She didn’t understand that it had nothing to do with her, but rather the nature of her father’s work as a doctor.

“Call waiting” is the name of a telephone feature. If a third person tries to reach you by telephone while you are already using it, it gives a little beeping noise to alert you that someone else is trying to call. You can ignore the beep, or you can answer the second caller. If you answer the beep, you can put the original caller on “hold” while you speak to the other person.

If you understand the above, you can probably figure out that she learned to answer the call waiting beep because it could be an important caller advising of an emergency that her father needed to deal with. They took two cars so they could get home (in one car) and dad could get to work (in the other) if he received a call while they were out.

"She used to take it personally" as she got sad when it first started happening and she realized it, she didn't understand as a child why he would have to leave in the middle of things, her mom had to explain to her why it kept happening. The last sentence means that she got used to hearing her dad's phone beep meaning there was an emergency thus he had to leave.

Thanks!