Translation:It is important that she look around.
This is a manifestation of the subjunctive in English – much more common in writing and relatively unheard in spoken language. Most verbs in the subjunctive take the indicative form, and so you see unusual (but correct) constructions like this one. I would say grasping this concept in English is more important for farther advanced speakers than the equivalent concept in Romance languages.
Why does this include "si"? Is it reflexive or passive? if so, why? Would the sentence function equally well without "si"?
edit: I checked WordReference.com (http://www.wordreference.com/iten/guardarsi). It gives definitions of "to avoid" and "to look out (for someone)" for guardarsi. Before I was merely puzzled by DL's translation; now I'm baffled.
re-edit: OK. I checked other sources. Guardare intorno and guardarsi intorno both mean "to look around"; however, the latter seems to be used much more often that the former. This one http://it.bab.la/dizionario/italiano-inglese/guardare-intorno even has an example with a penguin. Maybe that explains why penguins are in so many of DL's early exercises.
"Guardare attorno" and "guardarsi attorno" could be translated in the same way in English, but they have a distinctive difference in meaning in Italian. "Guardare attorno" means just to look around for something, but an Italian would just use other verbs instead of "guardare" in this case ("cercare" for example, maybe adding in a locative "qui" or "lì", or "dare un'occhiata" - dialectal though). "Guardarsi attorno" is much more vast: it could mean "to take a look around" literally, "to look out for oneself" as you wrote, but in a different context I could prefer "checking one's possibilities in the immediate vicinity" (both temporal and spatial). It's important to note though that in "guardarsi attorno" meaning "to look around", the "si" is neither reflexive nor passive: in this case "si" is bound to "attorno", meaning "attorno a sé". Hope this clears up some doubts.
This comment thread seems to get more attention than others that I follow. So if anyone is interested in reading a bit more about why "look" is correct:
Money quote: "This version uses the present subjunctive. The present subjunctive in this construction was once less usual in British than in American English; but in the last twenty years or so, it has become more common (perhaps because of the increasing daily contact by email, etc. between Americans and British speakers). It seems the most formal version: I would be surprised if a member of my family said it to me, but less so if a headmaster said it about a pupil.
When I say "The present subjunctive in this construction was once less usual in British than in American English", I mean "it was less usual in the early to mid 20th century". In pre-19th century English, the present subjunctive was much more common."
Actually we have 2 sentences in these cases, the main and the secondary one, that is introduced by a che. I cannot understand when we translate the italian subjuctive to a subjunctive (extinct or not???) in English and a pronoun introduced with a preposition, to for instance, or without, as an object to the main phrase and an infinitive. A rule of thumb?
I believe the word that it here is an Object in the Relative Clauses. Therefore, it can be omitted. Please be illustrated as to when that can be omitted. It's common to drop 'that' when it is the object of the relative clause it introduces. https://www.thoughtco.com/uses-of-that-1210017