- Consider first the case of a manifold $M$ seen as the space of units of the groupoid structure $M\to M$ where $s=t=id_M$ is the projection and no pairs are composable, so that you have only identities. Then the corresponding groupoid $C^*$-algebra is nothing but the standard $C^*$-algebra of continuous functions on $M$.
- Take now $G$ to be a (Lie) group seen as a groupoid over a point. Then the groupoid $C^*$-algebra is the group algebra of $G$ which can be seen as a completion of the universal enveloping algebras of the Lie algebra $\mathfrak g=Lie(G)$.
Generally speaking the groupoid $C^*$ algebra can be considered as a mixture of this two aspects: continuous functions on units plus additional informations on group of symmetries resting over orbits. Examples that can be thought of this kind: orbifolds (can be seen as groupoids), group actions, foliations. To each such object you may associate a Lie groupoid and a corresponding $C^*$-algebra which carries informations on orbits (points of the orbifold,orbits of the action, leaves) and their internal symmetries (discrete group over singular points, isotropy of orbits, holonomy of leaves). This is why the groupoid $C^*$-algebra is sometimes referred to as a desingularization of a singular space, such as the space of leaves or the space of orbits.
To any topological groupoid + a choice of Haar system you can build a groupoid $C^*$-algebra. If your groupoid is Lie you basically have a canonical choice of Haar system essentially unique.
There is a very nice paper by Ieke Moerdijk on Orbifolds as groupoids that you can easily find online.
Also searching for "foliation C^* algebra" will provide you way too many refs. One I like is the following