"Sich umsehen" (to look around) is a reflexive verb.
Like French, German has many reflexive verbs, whereas the corresponding English verbs are often not reflexive. The reflexive pronoun (myself, themselves, etc.) is an integral part of the verb and can't be translated literally into English.
Ex. to hurry - sich beeilen
They have to hurry. = Sie müssen SICH beeilen (literally: They have to hurry themselves; real translation: They have to hurry.)
As the reflexive pronoun is an integral part of this verb, you can't say "Sie müssen beeilen" - that would be ungrammatical.
It's the same with "sich umsehen" - you can't say "Sie sehen um" - that doesn't make any sense in German because "sich umsehen" is a reflexive verb and the reflexive pronoun can't be omitted.
Is it more nuanced than this though, since there are three categories:
1) True reflexive verbs
2) Partially reflexive verbs
3) False reflexive verbs
Where 1) are as you described, 2) can take on different meanings if used reflexively or not (but either is valid), and 3) are used reflexively in some situations but not in others, and yet retain the same meaning.
What I found is that German constructions necessitate an object - it's incomplete or there's something wrong if one isn't clearly defined.
I think it is understandable that the English "They look around" might sound incomplete to the ears of non-English speakers (although it's fine in English) as in they look around...what?/around where?
Well, I suppose the English sentence isn't complete either so you might want to say: "They look/are looking around for a room" but other than providing such a direct object -room-, you might just want to indicate that they are looking around for the sake of looking around (for their own purpose may that be for pleasure/interest or whatever...they're willing to look around themselves - first hand), so when you don't have an object, one means that: "They are looking around for themselves." or as what can actually be often used: They are having a look around themselves :- the reflexive sort of filling an object position. It is implied in English but German necessitates it.
Another way of looking at it is that sehen is reserved to mean to see as in to register (in your mind) like an instrument as opposed to schauen or ansehen:
to see (something/somebody) as (whatever) = (etwas/jemandem/n) (_) sehen
e.g. etwas optimistisch sehen = to see something as optimistic = to be optimistic about something...you can't simply sehen um as you need an etwas/jemanden element... in here, it is the reflexive that is the jemanden.
It's not always logical perhaps. As the other example given here in one of the replies, it would be "Sie mussen beeilen sich" as opposed to just: "They must hurry." To a German ear, I think the latter (English sentence) makes it sound like it's not complete: "They must hurry what?" as if hurry is an action verb that takes an object with it .. I suppose you would do that in English oftentimes: Hurry with the cooking (hurry cooking); in that regard, "They must hurry themselves" makes sense.
It is easier, I suppose, to just accept which verbs are reflexive and habituate yourself (no pun intended) into using those like so.
I don't think English is incomplete, the language native to me entirely lacks inflections, and uses reflection the same way as English, and it works just fine. The only way I can understand things like this is to think that these German verbs are used differently, "sehen um" doesn't just mean "look around", it means both "look and rotate themselves around", because they need to do so to make themselves look around. Of course this is just my opinion, but I really can't make sense out of it otherwise.
The problem is that English speakers are trying to understand German as they understand English. Mostly all reflexives in German are also reflexives in Spanish and the verbs change of meaning or are senseless without the reflexive particle. I feel that English reflexives are artificial with the -self particles. They are not real reflexives. We also have those words in Spanish (-mismo) and they sound unnatural if we use them as reflexives. So bad that modern English is just a shadow of what old English was, even with the enormous contributions that Shakespeare made to improve the language.