It's like an idiomatic expression "guardarsi intorno" = to look around; there are a lot of terms using "guardarsi".
"guardarsi" itself means "to look at" or "to take care"
guardarsi intorno = to look around
guardarsi attorno = to keep yourself informed; but also: to be careful
guardarsi alle spalle = to be on guard; to be wary
You are on the right track..I believe. Verbs often have a subject and an object. I look at the mirror....Clearly ( I ) is the subject and (the mirror) is the object.( Guardo allo specchio). Here the verb is clearly not reflexive. If however the subject and the object of the verb are the same it becomes reflexive. I look around (I look around myself). The verb Guardare now becomes the reflexive verb Guardarsi. Mi guardo....Ti guardi etc
Maybe this helps. Some verbs simply are reflexive, without necessarily having an english equivalent reflexive construction. Other languages are similar. For example "to catch a cold" in German is reflexive while in english it isn't. It doesn't always make sense why it's reflexive in one language and not in another. It's just something to learn.
My understanding of it is "guardare" is used as a transitive verb, meaning one that takes a direct object -- one looks AT something. "Guardarsi" I think is used more in the sense of "looking around". I know that doesn't explain WHY it's reflexive, other than to say that in some languages (German e.g.which I'm much more familiar with) some reflexive verbs simply have to be used reflexively and don't necessarily have an English (reflexive) equivalent. So, e.g. "to catch a cold" in German is reflexive. Italian has the same sort of verbs which may always be reflexive, with or without an equivalent English translation, or they may be sometimes reflexive, sometimes not - "guardare/guardarsi" in which case their meaning and use changes, maybe slightly, maybe a lot. Hope this helps.
Guardare means to guard/to watch/to look at and it needs an object, - what is being watched/looked at. In this sentence the object really is "around myself". So here "I" am both the subject (the one who watches) and the object (the one who is being watched around). This kind of construction is called reflexive and requires the use of "mi" (~myself).
Mi = myself
sono guardato/a = I watched / looked at (m./f.)
intorno = around.
Myself, I watched, around ~
I watched around myself ~
I looked around
From the use of "guardata" we also know that it was a woman that was "looking around".