Both sentences are grammatical, but mean slightly different things.
I am studying abroad in the United State from this summer.
This means you will start studying in the future ("this summer").
I have been studying in the United State since this summer.
This means you started studying this summer and are currently studying
In English, the progressive or continuous forms don't only mean a specific action in progress (e.g. "I am walking to school"). They can be used for broader recurring activities, sorta like viewing an action over a longer time frame.
I am reading a long novel
This could either mean a) the speaker is currently reading now or b) the speaker is in the middle of the process of reading the novel (i.e. started it but hasn't finished) even if they are not currently reading at this moment.
Verbs relating to activities can often be used that way, including studying