The Friday holds importance in Islam for many reasons. We call Friday Al-Jumu'ah because it is derived from Al-Jam', literally meaning "gathering". That is when Muslims gather in places of worship for prayer, weekly. Following are some of the important events associated with Friday:
It was a Friday when Allah finished the creation, with the creation of Adam (alayhi salam).
Ironically, it was a Friday when Adam (alayhi salam) was taken out of the heaven.
It will be on a Friday that the Last Hour will commence.
Apart from this, there is a hour on Friday when if a Muslim prays and asks Allah for something, He will definitely meet his demands.
To answer the question, why Friday was chosen for prayer specifically for Muslims- We believe that Friday was the chosen day for prayer also for the other nations before Muslims i.e. for the nation of Moses (alayhi salam) and the nation of Jesus (alayhi salam). But they were led astray from it.
It is narrated by Abu Huraira that the Messenger of Allah (may peace be upon him) said:
It was Friday from which Allah diverted those who were before us. For
the Jews (the day set aside for prayer) was Sabt (Saturday), and for
the Christians it was Sunday. And Allah turned towards us and guided
us to Friday (as the day of prayer) for us. In fact, He (Allah) made
Friday, Saturday and Sunday (as days of prayer). In this order would
they (Jews and Christians) come after us on the Day of Resurrection.
We are the last of (the Ummahs) among the people in this world and the
first among the created to be judged on the Day of Resurrection. In
one narration it is: ', to be judged among them".
(Sahih Muslim : Book 7, Hadith 32)