A lot of irreligious people don't really understand the purpose of traditional rituals and ceremonies. But then, neither do a lot of religious people. Some might think the purpose is simply because God wills it. But even if that were true, the explanation of God sure isn't instructive. It's really hard to say why God wills it. And since you don't know why, you know very little about the exceptions, when ritual and tradition are no longer worthwhile.
So what is the purpose of ritual? Sometimes, there is a functional purpose. For instance, every time I practice my flute, I warm up first. I play long tones, scales, vibrato exercises, etc. I don't really know how or why, but I believe that these warm-ups improve my subsequent practice session, and have long term benefits to my technique. But I certainly don't have any scientific papers to establish these facts; as far as I'm concerned, it is simply folk wisdom taught by my teacher. And that's fine. The purpose of this ritual is to apply wisdom which has been accumulated over however-many generations, even if I don't fully understand the benefits.
But if that seemed like a strained example of a ritual, that's because it is. After all, it doesn't really matter how exactly I warm-up. It doesn't really matter whether if I play the scales in chromatic order, or in order of fifths or fourths, or if I decide to play harmonic minor scales instead. I could skip it altogether, and all that happens is I sound like I haven't warmed up. What matters is the functionality of the warm-up, rather than the warm-up itself.
But the essence of a ritual is not in its functionality, whatever that may be. The essence of a ritual is in the order it imposes in our chaotic lives. For instance if you want to marry someone, you might be confused about how to propose. I know I would be nervous. I would worry about all the details, about any little thing that might go wrong. Luckily, there is a standard format for a proposal which everyone knows about. You drop down on one knee, and ask, "Will you marry me?" Above and beyond that, you may add whatever you like.
The same goes for the marriage itself. If marriage ceremonies didn't exist, I would feel kind of weird about just saying, "Okay, I guess we're married from tomorrow forward. We'll be married... after midnight?" It would feel as if nothing important had happened. Like we could just go back to how it was in an instant. But wasn't it important? Shouldn't it have felt more special?
I could go on and talk about other ceremonies marking important moments of our lives, such as baptism, confirmation, or funerals. But there are also lots of rituals which are performed on a more regular basis, such as a weekly church service.* The reasons for these regular rituals is much the same. But rather than providing a structure and format for your entire life, they provide a structure and format for your daily life. If, for example, you're a church-going Catholic, you go to church because it gives you some comfort in your chaotic life. In that brief service, you know who you are and what you are doing--you are a Catholic, and you are participating in a ritual with a prescribed role. You also know what to feel--perhaps it is a sense of devotion, or a feeling of "I'm going to go out there and do good this week!"
*Please note that weekly services are not entirely ritualistic, but also serve several functional purposes.
Or maybe you feel none of this. Maybe you feel even more confused, or maybe you find the whole thing positively boring. I daresay that many religious people practice their religion not because they are getting anything out of the rituals, but because they simply feel obligated to do it. Well, here's what I think. Regular rituals are not for everyone. Nor are ceremonies. If a conscious and benevolent god existed, it would not want everyone to be forced into rituals, not even the subset of people who believe in him.