Does "they get bitter“ mean ”they become angry"?

Close. You can certainly interchange "get" and "become" (although "get" is a little more natural).

"Bitter" is close to "angry", but it is a slightly different emotion. Angry people tend to take action and demonstrate they are upset. Bitter people are cold and passive.

The word "bitter" does not suggest that the individual is "angry". An anger response is usually an immediate response, with physically or emotionally visible results) ( namely a rise in blood pressure, a redness of the face, a significant change in disposition).

A bitter person may also be called a "sour" person. Namely, not having a "sweet" disposition, but rather one that is more "caustic". A bitter person is usually one, who would like to take action in some form or other, and usually does.

A mass killing, recently took place on a college campus in California. The 20 yr old killer was bitter, that girls didn't like him and so to "get even" he went out and murdered/injured many individuals using both a knife and gun as weapons.

This killer had been bitter for about 3 years. He waited, the bitterness in him increased over time, and then he took action in the form of murder and destruction.

Obama said,
"Hold on one second, sweetie."

I've heard Obama call some females "sweetie" on other occasions. I've never considered the term to be offensive, when Obama used it. It is a bit surprising however, that Obama used this word, since it is mainly used by much older men and perhaps more often at an earlier time in the history of the USA.

Likewise, the word is used more often in certain parts of the US, as for example, it's more common in the South, than in the New England area.

This is because New Englanders, for the most part, have rather cold dispositions, as opposed to those individuals living in the South where personalities tend to more warm, less "bitter" and definitely more outward going.

"Bitter" does not mean "cold". You can be "cold" without being bitter.

If you are "bitter" that means you deeply resent something you feel is unjust. Cold simply means "without outward emotion".

For that matter, someone from the South can put be "warm", meaning expressive of outward emotion, but still be "bitter" inside (i.e. deeply resentful in private).