The following verse says how our world (there are other worlds that were created by Allah but they are irreverent to us) was created:
Have those who disbelieved not considered that the heavens and the
earth were a joined entity, and We separated them and made from water
every living thing? Then will they not believe? -- Qur'an 21:30
Ibn Khatir (a highly respected Muslim scholar of 13th century, he was born in 1301 and died in 1373 AD) explained the verse as follows:
(Have not those who disbelieve known) means, those who deny His Divine
nature and worship others instead of Him, do they not realize that
Allah is the One Who is Independent in His powers of creation and is
running the affairs of all things with absolute power So how can it be
appropriate to worship anything else beside Him or to associate others
in worship with Him Do they not see that the heavens and the earth
were joined together, i.e. in the beginning they were all one piece, ...
(And We have made from water every living thing. Will they not then
believe) meaning, they see with their own eyes how creation develops
step by step. All of that is proof of the existence of the Creator Who
is in control of all things and is able to do whatever He wills.
Reference: Tafsir Ibn Khatir
Many Muslims scholars have interpreted "a joined entity" as a reference to the big bang. Most Muslim scholars agree that creation develops step by step but a lot of them reject human evolution and say that humans were directly descended to earth from heaven. Some other scholars disagree with this view and say humans also evolved from other species. Ibn Khaldun (Islamic scholar, historian and scientist, Born: May 27, 1332 AD) said:
The animal world then widens, its species become numerous, and, in a
gradual process of creation, it finally leads to man, who is able to
think and to reflect. The higher stage of man is reached from the
world of the monkeys, in which both sagacity and perception are found,
but which has not reached the stage of actual reflection and thinking.
At this point we come to the first stage of man after (the world of
monkeys). Reference: The Muqaddimah, Chapter I