Hello everybody, ! ¶
My name is Luca and I'm an economistcs student.

The subject is called economics, and a person who does that professionally is an economist. Also be aware that the pronunciation is different for those two: ecoNOMics versus eCONomist

I am atin my second year of a Master's degree in Business Administration.

"I am in the second year" would also be fine. You could even say "I am a second year economics student, working towards a masters in business administration."

I want to improve my language businessbusiness language skills (I want to reach B2 level as soon as possible XD) so that is why I decided to enrol in the Erasmus+ programme and cominge here, into Frankfurt, where I'm writing this text.

"so that is why I decided" is not wrong, but "that is why" is unnecessary in the context of this sentence, because it is clear that the second half (enrolling in Erasmus) is because of the first half (improving language skills). If you had written more (for example 2-3 sentences) about wanting to improve your language, you could use "That is why" then, to make the connection clearer.

Firstly, because of sStudying abroad means a lot forto me, starting from finding because I can learn about new cultures and different viewpoints from people comingfrom all over the world.

"Firstly because" makes this seem like a continuation of your previous explanation (I enrolled in Erasmus to improve my business language skills), but this sentence is talking about something a little different. "Views" and "viewpoints" are almost interchangeable, but "views" is a more general word. For instance, a hotel can have good views from most rooms (ie you can see the mountains). Viewpoints is less ambiguous. "Find out" tends to mean "learn that something exists or has happened". You find out the score in last night's football match. "Learn" is better suited when you're talking about new cultures.

I think this programme is the bestoffers a fantastic opportunity to opbroaden your mind.

"Offers an opporunity" is a very common verb+noun combination in English. It's more formal. "Open your mind" is more casual and tends to have a negative meaning: you're saying that the person is close-minded, which means they are distrustful or not interested in new ideas. "Broaden your mind" has no negative meaning. If you say "the best opportunity" it's more common in British English to specify something narrower than everything. For instance: "the best opportunity of all the options available to me", "the best opportunity I have heard of", "the best opportunity of all university programs". British people don't like to overexaggerate, especially not in more formal writing. For this reason, I suggested "a fantastic opportunity" instead.

I've been here since 1st September 1st and I've noticed the power of English, which is the only solutionway to communicate and understanding other people.

It's not alwayso easy getting away from your home but, you know, it's just matter of time...

"always easy" and "so easy" sound like they mean something different but in this exact usage "it's not always easy" they really mean the same

Since I'mve been here I've learnt a lot of new things, in generalfor instance, how to respect different views ofn the thingssame topic.

"Things" twice in one sentence sounds a bit clumsy, so I changed it to "topic". This is a situation where the meaning of "views" is totally clear so no need to use "viewpoints". I added "same" to "different views on the same topic" because otherwise it sounds like you're refering to a specific topic that you have already mentioned. You don't need to make "topic" plural here; it will be understood that you mean "lots of times people had different views on a topic"

In conclusion, I'm appreciatinge thise opportunity themy home university gave to me, and I want to enjoy it until the very last day.

You could say "I appreciate this opportunity" but when you give more info on the opportunity ("my home university gave me") it's more common to say "the" rather than "this". "The very last X" is a common phrasing when you want to emphasise something and it seemed to fit your wish to enjoy the remainder of your time in Erasmus.

Hello everybody, ¶
My name is Luca and I'm an economists student.

I am atin the second year of a Master's degree in Business Administration.

I want to improve my language business skills (I want to reach B2 level as soon as possible XD) so that, which is why I decided to enrol in the Erasmus+ programme and cominge here, into Frankfurt, where I'm writing this text.

This is not the only reason I decided comingto come here.

Given that in the next sentence you give another reason for going to Frankfurt, i think you might have wanted to say "not only".

Firstly, because of sStudying abroad means a lot forto me, starting fromas I can finding out about new cultures and the different view froms of people coming from all over the world.

It reads better if you don't use "firstly, because" here.

I think this programme is the besa great opportunity for me to open myour mind.

I've been here since 1stthe first of September and I've noticed the power of English, which is the only solution toway I can communicate and understanding other people.

It's not so easy gettbeing away from your home but, you know, it's just matter of time...

Since I'mve been here, I've learnt a lot of new things, in general,particular how to respect people's differenting views of then things.

In conclusion, I'm really appreciatinge this opportunity themy home university gave to me, and I want to enjoy it until the very last day.

Using "very" makes it sound more final.