Sitting in seiza while Sensei or a sempai is demonstrating is not only proper etiquette (to show respect for "those who came before"), but is also essential if you don't want to get injured accidentally standing too close to the "demonstration area".
You may sit cross-legged too, especially if you're injured or feeling uncomfortable, but you cannot sit with your legs outstretched.
It's also improper to stand up while Sensei is sitting; should you have to move across the dojo, only shikko is acceptable.
Naturally, these are the rules at my dojo, I wouldn't know if other dojos are more lineant.
As for the meaning behind sitting in seiza (which of course isn't proper to Aikido alone)...well, seiza is the formal Japanese sitting posture, not only in Budo but also in arts and crafts. "Sei" means "proper, true" and, yep you've guessed it, "za" means "sitting posture".
Proper seiza helps to naturally align your body and spinal column.