The word "being" is serving as what's called a present participle in these examples. It is helping to create a form of the verb called the present continuous tense.
You are probably familiar with a variety of tenses, but just in case you don't remember their names, here are some refresher examples, for the verb "feed". I chose an ordinary transitive verb (one which takes an object, i.e. one in which someone does the verb to something) to show the effect of different tenses more clearly:
(Simple) Perfect I fed the baby
Pluperfect I had fed the baby
Imperfect I was _feeding_ the baby
(Simple) Present I feed the baby
Present Continuous I am _feeding_ the baby
Future I will feed the baby
Future Perfect I will have fed the baby
Note that 2 of these, the Imperfect and the Present Continuous, use the present participle form of the verb itself, "feeding".
Now, suppose that you are the baby. Instead of the active form of the verb, "to feed", we can switch to the passive form, "to be fed". Then, the sentences from earlier in these two tenses change to:
Imperfect I was _being_ fed
Present Continuous I am _being_ fed
So, the word "being" shows up most commonly in these two tenses, in the passive form of the verb.
Note: The verb "to be" is an additional, unique case where the word "being" can appear to form these same tenses, but the verb is intransitive. For example:
Imperfect I was _being_ careful
Present Continuous I am _being_ careful
Modern American English relies heavily on present continuous tense verb forms instead of the more mundane present tense. In fact, the simple present tense is only regularly used in one way - to reflect an ongoing, regular activity. A few examples:
"How do you get to work?" "I ride the bus."
"What kind of work do you do?" "I feed animals at the zoo."
The normal and more common form of the present tense in AmE is the present continuous:
"I'm watching my weight".
"He's being asked to sign the papers."
"She's getting really upset."
BritE does the same thing, but the simple present shows up a bit more, just as a variation in usage.
Note: There is one rather uncommon usage of the word "being". It sometimes appears to start an adjectival phrase, which is a phrase that describes something. For example:
"The ship's captain, being taller than the rest of the crew, was able to reach the swinging cask."
There is another occasional, but incorrect, usage where the word "being" is used as part of a kind of additional dangling phrase that's trying to do the job of a conjunction:
"I ate the rest of the cake, the reason being that I was depressed."
These are really two separate ideas, and could be correctly framed by inserting the conjunction "because" and dropping the awkward phrase "the reason being that":
"I ate the rest of the cake because I was depressed."