There are several useful points in the comments, but I want to go beyond them and try to give a more comprehensive answer, so this question doesn't linger unanswered. Some great sources are May's Geometry of Iterated Loop Spaces (GILS) and A Concise Course in Algebraic Topology (CCAT). As the OP points out, if $X$ is a loop space, then concatenation makes $\pi_0(X)$ a group.
Is this the only motivation behind the study of loop spaces? Or what other interesting aspects can be gained from studying loop spaces from viewpoint of homotopy theory?
No, there is much more to loop spaces than the observation that concatenation yields a group structure on $\pi_0(X)$. First of all, motivation is easy. Since homotopy theory is intimately tied to paths, loop spaces themselves are fundamental objects, e.g., because they allow you to shift dimension as $\pi_i(\Omega X)\cong \pi_{i+1}(X)$. The operation of taking loops is connected to the suspension operation as follows. Let $X$ and $Y$ denote based spaces, $F(X,Y)$ denote the space of based maps between them (so $\Omega X = F(S^1,X)$), and $\Sigma X = X\wedge S^1$ denote the suspension of $X$. The usual hom-tensor adjunction tells you that $F(\Sigma X,Y) \cong F(X,\Omega Y)$. Taking $\pi_0$, we have $[\Sigma X,Y] \cong [X,\Omega Y]$, and composition of loops turns this set into a group. Hence, $\Omega Y$ is a cogroup object in the homotopy category of pointed spaces, and this is used in Hovey's book to set up the much more general homotopy theory encoded by model categories.
Since the theory of $\Omega$-spectra starts with a sequence of based spaces $T_n$ and weak equivalences $T_n\to \Omega T_{n+1}$, loop spaces are also foundational to stable homotopy theory. They pop up in the long exact sequences induced by fiber and cofiber sequences, that allow us to compute things in stable homotopy theory. They pop up in Postnikov towers and Brown representability (since $K(A,n) = \Omega K(A,n+1)$). One way to prove Bott periodicity is to study the homotopy equivalence of $H$-spaces $\beta: BU \times \mathbb{Z} \to \Omega^2(BU\times \mathbb{Z})$. So, there's plenty to motivate the study of loop spaces. Let's say more about "interesting aspects."
The introduction to Adams' book Infinite Loop Spaces mentions work of Morse and Serre computing the number of geodesics on a Riemannian manifold using loop spaces, work of Serre on $H$-spaces and the Pontryagin product on $H_*(X)$, and the development of the Leray-Serre spectral sequence and its resulting homology calculations. Loop spaces give us more to compute with, and double loop spaces, $n$-fold loop spaces, and infinite loop spaces, give us even more.
As has been pointed out in the comments, the recognition principle says that $n$-fold loop spaces $\Omega^n Y$ are (up to homotopy) the same thing as $E_n$-algebras. For $n=1$, these are the same as $A_\infty$-spaces, as discussed in the preface to GILS. The Moore path space trick Naruki mentioned (parametrizing loops by $[0,t)$) gives a model for a strictly associative and unital topological monoid of loops, and the usual loop space is a deformation-retract, which is one way to understand the $A_\infty$-space structure. It's easy to show that $\Omega Y$ is a grouplike homotopy associative $H$-space, but the $A_\infty$-structure is better.
As pointed out in the link Najib provided, the $i^{th}$ stable homotopy group of $X$ is equal to $\pi_{i+k} \Sigma^{k} X = \pi_i \Omega^k \Sigma^k X$ for sufficiently large $k$, so spaces of the form $\Omega^k \Sigma^k X$ for $1\leq k \leq \infty$ contain a tremendous amount of information about $X$. As pointed out in the preface to GILS, this leads you naturally to the James construction and to Dyer-Lashof operations, which are essential for understanding the algebraic structure of the (co)homology of $X$, for an understanding of power operations, and for computations in the Adams spectral sequence. In GILS, May finds geometric approximations to these spaces, and descriptions of $H_*(\Omega^n \Sigma^n X)$ as functors of $H_*(X)$. The resulting understanding of Dyer-Lashof operations is the foundation upon which much computational work has been done, as wonderfully summarized in an article of Tyler Lawson.
