This is my interpretation of the arrangement. As a relative newcomer to D&D, I may have gotten some lore stuff wrong, so feel free to correct me.
From my reading of it, Toril is a world where the touch of the gods can be felt in all aspects of life. Their hands and their wills are seen behind every stroke of fortune and misfortune. A bountiful harvest is not mere luck or a farmer's wisdom, but Chauntea's blessing. Hurricanes and tsunamis are not coincidences of weather but a sign that Umberlee's wrath has reached its apex. The sun rises every day and shines upon the land because Amauanator makes it so. A prospering business owes its success to Waukeen and Tymora's presence, and Basheba's absence.
As such, unless you're an acolyte to a specific deity and are involved in clerical duties, the line between religion and daily life is... nonexistent. For the people of Faerun, they owe their lives and the world to the gods. Divine influence on their culture is pervasive. You go to work? The gods are watching. You come home to your family? It is the gods' blessing that your family is prospering. Your son has found a lover and is thinking of getting married? Sune had a hand in that. Your daughter has saved enough money to go to university? Oghma and Deneir will be taking interest in her studies. Your rival was interfering with your business but suddenly died of a heart attack? That was the hand of Hoar. And in some cases, that is not mere metaphor but actual, conscious divine intervention, and it has happened enough times that people have begun to believe that it is the rule rather than the exception. For in many cases, if a god is incapacitated, or neglects their duties, the rules of reality around their domain begin to fall apart. Tymora bites it? No more luck or good fortune. Auril disappears? No more winter, and summers become more intense.
So somebody who refuses to even give lip service to any of the gods, whether out of spite or for some other reason, would be seen as equivalent to denying the laws of physics. They'd seem as crazy and deviant as, I don't know, let's say an anti-vaxxer would in real life. And I don't know about you, but a lot of people I know really wish those people could get shut up in a wall. Or just imagine that you went back to Athens at its golden age and started yelling about how the Olympians are gluttonous tyrants and rapists that care not for the mortal man. You might be right, but you're gonna end up denied service at most respectable institutions at best, and outright mobbed at worst, if only out of fear that your blaspheming will cause a sudden chance of thunderstorms. Just ask Socrates.
I come from a really religious household, and while I'm only going to church as lip service nowadays, there's still enough in my upbringing left to grok how a religious cosmology like Toril's could be seen as remotely acceptable. I mean, is there any real difference between "believe in and honour God for it is God's will and if you repudiate him you go to hell" and "believe in and honour the gods for it is by their hands that the world turns and if you repudiate them you go to the Wall of the Faithless"? Hell, in the latter scenario, you have an all-you-can-eat plethora of gods to choose from, and will probably unconsciously end up with one of them due to the fact that pretty much every trade in the land has a patron deity that you'll be praying to to bless your work or crying out to in desperation if your business comes crashing down. If you were a good person - or even a bad person - in life who at the very least followed the ideals of their trade, there will probably be at least one god who will vouch for you when you're standing in front of Kelemvor's judgment seat.