This is a large, complicated question so for a truly satisfying answer I suggest you simply consult a book. My recommendation is Dale T. Irvin and Scott W. Sunquist's History of the World Christian Movement: Earliest Christianity to 1453. You can get the kindle version for 15 dollars (unless you're boycotting Amazon of course). I'm going to give you a quick run through of the political events leading up to Christianity becoming the state religion.

There was not a smooth "switch" from polytheism to monotheism. Politically, Emperor Constantine I was the first to recognize Christianity as a legitimate religion, rather than a pernicious cult, with his 313 Edict of Milan. The Edict extended the ability to freely practice Christianity in public, among other religions, and de jure ended Roman persecution of Christians. Constantine's reasons for this are complicated, but the common story is that during his struggle to win a civil war against Maxentius he turned to the Christian God for help, in the same way another Roman might turn to Mars or Minerva for help in war. He attributed his subsequent victories to the Christian God, and became favorable towards Christians. However, Constantine I did was not baptized until shortly before his death in 337, and it is arguable whether or not he was ever a Christian in the traditional sense. It seems that he simply adopted the Christian God because he viewed Him as the stronger than the traditional Hellenistic deities.

Christianity would still not become the state's religion until 380, when Theodosius I issued the Edict of Thessalonica, declaring that all subjects of the Empire were to profess faith in Nicene Christianity. Until this point, the older Hellenistic religions, various forms of Christianity, and other minor religions all co-existed within the Empire. Theodosius reversed earlier Christian persecutions, and began a campaign to systemically demolish rival Christian groups, specifically Arianism, and the Hellenistic religions.

There was not a smooth transition from Constantine to Theodosius. The Emperors in between had their own religious views. Constantius II preferred Arianism to Nicene (Orthodox) Christianity. Julian rejected Christianity entirely, renewed the persecution of Christians, and attempted to revive the old Hellenistic religions through Neoplatonism. Julian's reign (361-363) is generally regarded as the last gasp of polytheism in the Roman Empire, but certainly not the last time polytheism would be practiced in Europe. As late as the fifth century, St. Augustine found the various pagan religions threatening enough to specifically address them with careful arguments in City of God. Unfortunately by this point the sources are dominated by Christian writers, so it is difficult to tell just how many people were practicing polytheistic religions and in what forms, but there were enough to seriously perturb Christian thinkers like Augustine.