It is fair question probably encountered by everyone who begin to study these things. 

The modern line of thought put everything into bigger picture, and that relation is some particular case of more general phenomena. But for that one needs to be familiar with certain ideas from several areas, which are kind of far from another, so probably it is some source of difficulties (at least for me when I started to study these things many years ago). 

Let me try to sketch the ideas along that line of thought: 

1) Representations of Lie group — more or less same as representations of Lie algebra. 

2) Representations of Lie algebra — same as representations of  universal enveloping algebra.

3) Universal enveloping algebra — is an associative algebra, so as for any algebra looking for representation we should pay attention on ideals,
(which are related to kernels of representations).  
So look on principal ideals  $\hat C_i = 0$ — for some elements.
But for non-commutative algebras $\hat C_i$ should be central elements for the ideal to be two-sided ideal and everything works smoothly. 

So we arrive to the simple idea that:  "Central Elements = Constants" — gives us some (actually main) source of the representations of any non-commutative algebra. 

4) The final step is to relate "Central Elements = Constants" to coadjoint orbits. That is better to understand in the general framework of quantization. 
But let us try to be a simple as possible. 

5) From $U(g)$  to $\operatorname{Fun}(g^*) = S(g)$ — by isomorphism of modules over $G$. 
The point is that universal enveloping algebra is kind of special algebra which is not far from commutative algebra of functions on the $g^*$. (It is its quantization, but let us try to avoid that term). 
Central elements in universal enveloping algebra (Casimir elements) they  are, of course, invariant with respect to action of the Lie group — but as representation of the Lie group universal enveloping algebra is isomorphic to $\operatorname{Fun}(g^*)=S(g)$ — just the commutative algebra. 

**Point:** So central elements in $U(g)$ are exactly the same as invariant elements in $\operatorname{Fun}(g^*)=S(g)$. 

6) That is more or less all what we need. 
From central elements $\hat C_i$  in $U(g)$ we got just functions $C_i$ on $g^*$. 

**Point:** $ C_i = c_i$ — define precisely the coadjoint orbits. Just because invariance of $C_i$ and so invariance of defining functions is more or less the same as the manifold to be an orbit. Since it is an orbit in $g^*$, it is a coadjoint orbit. 


 
So I hope the story above at least puts some ideas on the table and might be helpful.
It is not the end of the story — one needs to explain how to construct the representations. That story is somewhat open-ended — there is clear philosophy  why that might work, but it is not that much clear how to go from philosophy to constructions and when these constructions are possible, when not.
Also the line of thought above should be better understood in light of general ideas of quantization — where a coadjoint orbit is just an example of a Poisson leaf, and it is expected that Poisson leaves should give rise to representations — it is a part of general "classical" to "quantum" dictionary.