Etiquette and social manners allow us to practice safely as well as develop mutual respect and a supportive environment of awareness.
The martial arts are a discipline and physical technique is a vehicle for your personal refinement and spiritual growth. In addition to technique, how we conduct ourselves in the dojo and during practice is informed by etiquette.
Reigi (Etiquette / Manners)
One may encounter a lot of unfamiliar bowing and forms of etiquette. It's important to see the whole picture - the kind of personal development being encouraged, rather than each piece of etiquette or behavior. It is reasonable and expected to not memorize everything all at once. It takes time for a student, receiving occasional reminders and hints and paying attention, to gradually "get" the dojo culture.
As a general function, bowing and being mindful of posture, where one is facing, where one is standing, etc. serve to provide "checkpoints" to remember to be attentive, to practice mindfulness. It is up to the individual student to "mean it" and not just go through the motions. In this sense, one is in complete control of how one behaves - whether to do something deeply and sincerely or superficially.
There are numerous occasions on which to bow. As well as providing reminders to be attentive, they also often mark the beginning and end of something, whether it is practicing with a particular partner, entering a physical space, or "having the mic". Bowing brings into perspective that you are here in the dojo to work on yourself with partners who you support and who support you.
The Effect of Ritual
On the one hand, bowing, the particulars of how to sit, etc. are a part of the cultural tradition from which aikido was born. But we can ask ourselves, why have our predecessors thought to keep these details in our art?
There are various aspects of tradition that we could de-emphasize or omit. It's meaningful to continue including bowing as "checkpoints" to pause, however briefly, and take notice that something new is happening. Instead of a continuous stream of activity, we get a prompt to reset our minds and pay attention. One way to frame bowing when getting on the mat to children is to say that it is like going through a door into a special room (though there is no literal door or physical barrier). Bowing is a way to tell oneself, it's significant that 'now' I'm going into this space, I'm starting/ending this role.
In ancient cultures, when going to and coming home from war, rituals served as a kind of inoculation against trauma. We humans have a tendency to be fundamentally affected by the narrative we find, or place, ourselves in. Thus, framing our experiences, such as by ritual, changes how we come through them.