From various sources actually. Best of them are of course all the written texts by ancient Greek and Roman authors (and poets), in which we get told how the belief was portray. We are also able to interpret these texts to get additional information about rituals of general thought. Ever heard the modern saying that a book says a lot about the author that s/he isn't meaning to tell us? Well, it's the same with these texts. Take the very well known Iliad and Odyssey for instance. Homer - whether it was one person or several doesn't matter - isn't making any attempt to describe to the readers what kind of gods the characters are praying to, nor does he explain why they do a certain ritual for one god but not for the rest. That's the foundation of it.
What really gives us meat on the bones though are all the archaeological and epigraphical material around the Mediterranean world. Sometimes these archaeological finds can be compared to something an ancient author has written down, which can give sort of an eureka moment for an historian. More often though, the evidence is in the details. There are several settlements on Cyprus (just a random example that turned up in my head) where holes filled with jugs and bones give a testimony to the feasts and rituals there. I'm afraid the how that you're asking may take more than this one post to understand in this case, but I'll try to give a short TL;DR version of how it works.
See, when archaeologists dig they are very careful with how they did. You don't just stick a shovel in the ground somewhere and hope for the best. Instead of start of with a plan and then you dig trenches of various shapes and sizes. The shapes and sizes are usually determined by factors such as what you're looking for, what the budget looks like and the will of the dig leader. Anyway, back to what I was talking about before I rambled away: the jugs and bones. When you dig a trench like this you make sure to dig gradually downwards, meaning that you can create chronological layers (the more stops you do, the more detailed the chronology). So when we have sacrificial holes like these on Cyprus, we can sometimes tell the difference between a festival and what's merely an ordinary sacrifice. Essentially this is the ELI5 version of it, but hopefully it gives something of an idea towards how this process works.
Moving on to the parts that I'm actually concerned with, the archaeological artefacts and epigraphy. In my opinion, this is where the real stuff is. All of the above are very valid ways of interpreting and learning about the overall view of religion in the Greco-Roman world, but to learn how it really worked, you have to look at the details in the larger picture. Take Roman homes for starters, many of these would have had altars at home. We know that the richer families had these since, because they were made of stone, they have been preserved. The lower and poorer classes of the Roman world would often not have stone altars though. Yet they shared the same religion as the upper classes (the material shows this), which leads to the educated guess that they would have had wooden or perhaps clay altars to make sacrifices on.
Altars are in general very good sources/artefacts when it comes to religion in the Greco-Roman world. One perhaps very evident detail of it is that it shows organized paganism, as opposed to monotheism. Finer altars such as those located in temples or made as funerary stelae (gravestones) often also have epigraphical texts describing who or what the altar is for. In the former case the text on it can be a tribute to Jupiter Optimus Maximus for instance, showing clearly who the sacrifices on that particular altar would please. In the latter case, with the funerary stele, it shows another part of Roman religion to us.
That part is the household religion, which includes lares, the spirits of the ancestors. Lares are spoken of by ancient authors, but when we find funerary stelae we can get a much deeper understanding of how the religion worked. I just want to point out here that burial customs also play a huge part in our understanding of the lares. Here's an example of what the text on one of these stelae can look like:
[…] a hic s(ita) [e(st)]
[(L)]arus
matri^1
I'm not going to embarass myself with translating it because then someone else is going to come here and point out that my grammar is wrong! :) Anyway, what I meant to say with the epigraphical material is that they may as well be the holy book you speak of, except that they are scattered over a large area as inscriptions on different stones. It doesn't necessarily have to be funerary stelae speaking of lares, it can just as well be a small family altar that states the family's ties with Ceres, or similar. I think I'll stop there before I ramble on into oblivion. Greco-Roman religion is a rather complicated matter when it comes down to it and the big scholarly names are really heavy because of how well they know their field.
[1] Année Epigraphique, 1983, 496