RANDOM CHINESE LESSONS WITH FAN TIME
I KNEW someone would ask this sooner or later, i KNEW IT

BOTH are right, in my opinion.
很 (hen) = very.

however, it's a bit of a habit to chuck 很 in front of like... everything in chinese https://twitter.com/UndeadRobins/status/1248327683206938624

since 很 = very, it can be both an adjective and adverb, right?
for things like:
很开心 (hen kai xin) = very happy
很忙 (hen mang) = very busy,
the 'very' is put in front of words like this kind of more out of habit. we habitually prefer to say 'very __'

these can obviously be by themselves without the 'very' in front of them, or you can put other things infront, such as:
挺开心 (ting kaixin) = quite happy
比较忙 (bi jiao mang) = relatively busy
蛮开心 (man kai xin) = pretty happy
特别忙 (te bie mang) = particularly busy

you might ask "so in which contexts do you use those ones/when do you use 很 instead of those/when do you use which??"
my answer is: it's really up to you, just like in English, we switch up our adjectives/adverbs right? when would you use quite/pretty/very etc. in English??

anyways i got off topic. what i was trying to say is that verbs like happy/busy or adjectives can be by themselves, but more often than not, you'll see something stuck infront of it. that's just a habitual expression, to the point where it sounds weird by itself sometimes.

for your examples, 开心 and 忙 in particular sound really..... naked.... when they're said by themselves. We'd really rather not say them without something infront of it. Just think of it as the Chinese being v dramatic and exaggerating everything!

ok just to reiterate guys: DON'T say 开心/忙 etc. on their own.

DON'T say:
我开心 (wo kai xin)/我忙 (wo mang). you need to add 很 or one of the variations i mentioned above like:
我很开心 (wo hen kaixin) = i'm very happy
我很忙 (wo hen mang) = i'm very busy

you can't just think of it like in english where you say "i'm happy" and "i'm busy", in chinese that sounds WEIRD and that isn't done.

all the variations that i said in this thread above (and below) can be mixed up, used whenever, and not limited to 开心/忙. more in tweet below:

MORE variations:
我开心死了 (wo kai xin si le) = i'm happy to death = so happy that i could die
我忙得要命 (wo mang de yao ming) = i'm busy til it costs my life
我非常开心 (wo fei chang kai xin) = i'm extraordinarily happy
我忙着呢 (wo mang zhe ne) = i am being busy

so all of those above variations (there are probably more that i can't think of off the top of my head) can be switched around and used for other words too ok, i just used alternating 开心/忙 as examples but none of those are limited to those words. is it a bit more clear now?