The verb управлять is oblique transitive (косвенно-переходный).
That means that it can take a direct object, but the object is put in a case other than the accusative.
The direct object of this verb should be in the instrumental. That's how the verb works.
If you look at Russian cases (as well as cases in any other inflected language, for that matter), you'll see that while most of them do have a prevailing usage pattern (the accusative is mainly for objects, the dative is mainly for the recipients of an action etc.), this usage pattern is more like a general guideline rather than a strict rule.
In Latin, the dative can be used as dativus finalis, dativus ethicus, dativus auctoris and in all other kinds of roles which don't have anything to do with being on the business end of some action, despite the fact that it looks like this from the point of view of formal syntactic analysis. This is something you have to learn outright when you study Latin.
It's the same way in Russian; Russian verbs accept their arguments in a certain case and maybe with a certain preposition, which sometimes doesn't make sense.
When I was a kid, I was baffled when I first learned the expression ревновать кого-то (А) к кому-то (Б) ("to be jealous of someone (A) because of someone else (B)"). It didn't make sense to me right away. But as any other speaker, I learned how to use it and now it just sounds natural to me (unless I go over it again and again in my head until it doesn't).
Over time, verbs can change the way they govern their arguments. It used to be следовать кому-то, "to follow someone", and these days it's следовать за кем-то. It used to be скучать по ком-то "miss someone", in the prepositional, but now it's falling out of use in favor of скучать по кому-то, in the dative. In various Russian dialects, it's even скучать за кем-то. The latter is non-standard usage, but it's worthy enough of mention in almost every other style guide out there.
This exact thing happened with the verb управлять. It did use to govern a direct object in the accusative:
- И послаша о том бити челом ко благочестивому царю и государю нашему, чтоб государь их пожаловал: дал имъ в Казань своих бояр и правителей, кому их здержати и управляти.
- Бывшу собору на Москвѣ и избраша Коломенского епископа Геронтиа, яко достоина суща управляти богомъ поручное ему стадо.
- И аще така будеши вся своя дела управляти, то всяк тебя будет блажити и добрым человеком будеши слыти.
but then it switched the pattern and now it governs the instrumental.
The role the cases play in a sentence is not defined by a strict rule, although they do tend to follow patterns. When you learn a new verb, you also have to learn how it governs its arguments. Most of the time, it's intuitive, but sometimes it isn't.