From my understanding อยู่ means "to be at, stay or live" but I've noticed it being used at the end of a few sentences and it simply doesn't make sense to me / I can't figure out why it's used that way.

Here are two examples:

1. ผมกำลังรอใครบางคนอยู่ - I'm waiting for someone. (Could it eventually mean "I'm waiting here for someone"?)

2. ไม่เห็นหรือไงว่าผมยุ่งอยู่ไม่เห็นหรือไงว่าผมยุ่งอยู่ - Can't you see that I'm busy?

How should I understand อยู่ in these two examples?

Thanks.

ผมยุ่งอยู่

It shows the action is taking place right now.

I am busy right now.

In your first example, I would read it as, right now I am waiting for someone or I am waiting for someone just now.

Think of the normal Thai way of speaking,

Q. Yoo tee nai? where are you?

A. Yoo tee baan I am at home.

Q Tam arai yoo? what are you doing just now,

A. Doo tv yoo, watching tv just now.

ผมยุ่งอยู่

It shows the action is taking place right now.

I am busy right now.

In your first example, I would read it as, right now I am waiting for someone or I am waiting for someone just now.

Think of the normal Thai way of speaking,

Q. Yoo tee nai? where are you?

A. Yoo tee baan I am at home.

Q Tam arai yoo? what are you doing just now,

A. Doo tv yoo, watching tv just now.

First of all, thanks for taking the time to reply. I appreciate it.

I'm not trying to be smart or anything, but basically you're saying it's the "present continuous". Ok. If I look at my first example again, "กำลัง" (gam lang) is to show that the action, in this case "รอ" (the verb "to wait), is happening now. When I look at your explanation I would be using the present continuous twice in the same sentence... Now I'm really confused...

In Thai, กำลัง andอยู่ can happily exist together in the same sentence. Just a bit of emphasis of the 'present continuous'.

Although in spoken Thai (which always finds the shortest way), the use of อยู่ by itself seems to be most normal.

I accidentally stumbled across this on thai2english (am surprised I didn't find it earlier when I looked...). Here is how they describe the use of อยู่

"The "present continuous tense" is used for actions currently taking place, shown by to be + "-ing" in English e.g. "I am eating". In Thai, the pattern is กำลัง gam-lang + verb + อยู่ yùu although either of gam-lang or yùu may be dropped without affecting the meaning. This can be either in the present tense, or for describing events that were happening in the past."

Thanks to RickBradford and rgs2001uk for the input/explanation.