I want to understand why the concept of quasistatic processes is important and what problem would there be in the theory of thermodynamics if we don't define processes this way?
Thermodynamics (despite the name) is the study of systems in equilibrium. The set of all possible equilibrium states is called the state space. If you imagine a high-dimensional space, with a dimension for each thermodynamic variable (e.g., pressure, temperature, volume, etc), then the state space is a surface in that space determined by the constraints imposed by equation(s) of state. The ideal gas is a good example: consider an ideal gas with fixed particle number. We have a three-dimensional space with axes for pressure, temperature, and volume. But the equilibrium states of an ideal gas are only those values of $(p, V, T)$ that satisfy $pV = N k T$. The state space is therefore a two-dimensional surface. Almost all points in $(p, V, T)$ space do not represent an ideal gas. We could add another thermodynamic variable as a fourth dimension, e.g., the internal energy, $U$. But there is another equation of state: $U = \frac{3}{2} N kT$ (for a monatomic gas), so the state space remains two-dimensional.
A quasistatic process is a transition from one equibrium state to another equilibrium state where the system remains in equilibrium at all times. Therefore, a quasistatic process is represented as a curve on the state space, i.e., a continuous sequence of equilibrium states. This is obviously an idealization: such a transition would have to occur infinitely slowly. But, it is meant to represent the infinitesimal limit of making a very small change to the system, allowing it to return to equilibrium, and then repeating.
On the state space, we can apply the tools of differential geometry to calculate changes in the thermodynamic variables along a given curve (just as one does, e.g, on the curved spacetime of Einstein's general relativity). The basic tool for performing such analysis is the thermodynamic identity, which, in its simplest form, is:
$$
dU = T dS - p dV
$$
This is a relationship between covectors (or, differential forms) that is valid at every point on the state space. While the equation(s) of state determine what the state space is, the thermodynamic identity provides the fundamental relationship between thermodynamic variables on the state space. (Operationally, we can think of covectors as objects that can be integrated over a path).
All of this analysis is performed using quasistatic transitions/processes on a state space.
So, you might ask, why do we need to call them "quasistatic transitions"? Can't we just call them thermodynamic transitions, since they seem to be all there is? Well, for physical systems they are definitively not the only kind of transitions. As mentioned above, they are an idealization that is impossible to realize in practice. Most realistic thermodynamic transitions, from one equilibrium state to another equilibrium state, occur completely out of equilibrium, with only the initial and final states represented on the state space. Examples include: rapid expansion/compression of a gas, mixing of two gases, expansion of a gas into a vacuum, heating a pot of water on a stove (even slowly: the heating sets up a temperature gradient such that the water is always out of equilibrium). Unless we want to restrict the theory to only idealized, impossible processes, we must at least recognize that much of thermodynamics occurs completely "off of" the state space.
The preceding paragraph does not mean that quasistatic processes are not useful, however. Because each equilibrium state is represented on the state space, we could take a non-quasistatic transition (represented by just the distinct initial and final points on the state space) and consider a quasistatic transition (curve) that connects these two points. For any calculations involving changes in state variables (functions that depend only on the position on the state space, e.g., internal energy, entropy, and all thermodynamic variables) the result will be the same for the quasistatic and non-quasistatic process. A typical example is calculating the change in entropy when a hot cup of water and a cold cup of water are allowed to come to equilibrium with each other. The system is never in equilibrium while their temperatures are changing. But, because entropy is a state variable, we can use any quasistatic process that would connect the initial and final states, e.g., separately, and slowly, changing the temperatures of each cup using hot plates (to that final equilibrium temperature required by conservation of energy). In that way, the entropy change of each cup is can be calculated using the thermodynamic identity: $\Delta S = \int dU/T = \int c m dT/T$.
So, in summary: (1) quasistatic processes and the thermodynamic identity allow for calculations involving the integration over a path on the state space; (2) often these calculations can be used as a proxy for a process that is not quasistatic, but has the same initial and final states; (3) there would be no "problem ... in the theory of thermodynamics" if we consider other transitions, it's just that there are not formal tools for making calculations for them.
