Learn Russian: Не надо разбредаться, мы должны держаться вместе! - No need to wander off, we must stick together!

The base verbs of motion are брести and бродить (to wander, roam, or trudge). The prefix раз- combined with the reflexive particle -ся specifically indicates a group scattering or diverging in multiple directions. So, разбредаться means to wander off in different directions.

Why is разбредаться imperfective here? Could we use the perfective разбрестись?

You should use the imperfective here. When не надо means one shouldn't do something or there is no need to do it, it is almost always followed by an imperfective verb. Using the perfective разбрестись would sound unnatural.

Is the root of разбредаться related to the word бред (nonsense/delirium)?

Yes, they share the same root! The verb брести means to wander, trudge, or plod. The noun бред (nonsense or delirium) comes from the metaphorical idea of the mind wandering off its path.

Why does держаться have the -ся particle here instead of just being держать?

Держать means to hold an object. By adding the particle -ся, the action turns back on the subjects or describes a mutual action between them. So держаться вместе literally means to hold ourselves together, which translates naturally as to stick together.