Kendall Atkinson has done a big work in numerical solutions of integral equations. As with almost all other kinds of equations, the explicit solution of the integral equation is rarely known.

On the other hand, you shall ask yourself what is meant by the explicit solution. Usually, it means that the solution function $f$ of some integral equation $\mathcal I[f] = 0$ can be expressed via the finite number of algebraic operations involving some "known" functions, e.g. polynomials, exponentials and trigonometric functions. So that if you know that the solution is $f(x) = \sin x$, would you declare that it is explicit? You can barely compute the explicit value of $\sin x$ even if $x = 1$. Moreover, if you allow for a more general class of functions to appear in the expression of $f$, such as $\Gamma$-function or Bessel functions, perhaps you can solve explicitly a bigger class of equations. Finally, if you know that for some integral operator $\mathcal I$ it holds that $$ \mathcal I[f] = 0 \tag{1} $$ has a unique solution $\hat f$, the equation $(1)$ by itself serves as an unambiguous definition of the function $\hat f$. You can see now, that an explicit solvability of $(1)$ is quite a relative notion and it depends, in which class of functions are you looking for the expression of solution.

In the end, to answer your question about the real need for exact solutions, let me mention that it hinges upon the problem in hand.

1. Suppose, that the solution of $(1)$ describes some real phenomenon by itself. Then often numerical solution with precise bounds on the error is enough. Yet again, even $\sin 1$ can be computed in reality only with some imperfect precision.

2. However, if the solution of $(1)$ enters as a part of some problem, it is much more handy if you can express all the parts of this problem using the same language. Suppose, $f$ in $(1)$ further describes some dynamics $$ \dot x = f(x). \tag{2} $$ Even a small perturbation in value of $f$ can dramatically change the character of the asymptotic behaviour of $x$. Not to say, that the purely numerical expression of $f$ makes it hard to do the qualitative analysis of the ODE $(2)$.

I hope, it helps - though I am not sure whether this is the best place to ask such questions, as it may be not considered as a research-level question. Better try MSE

Kendall Atkinson has done a big work in numerical solutions of integral equations. As with almost all other kinds of equations, the explicit solution of the integral equation is rarely known.

On the other hand, you shall ask yourself what is meant by the explicit solution. Usually, it means that the solution function $f$ of some integral equation $\mathcal I[f] = 0$ can be expressed via the finite number of algebraic operations involving some "known" functions, e.g. polynomials, exponentials and trigonometric functions. So that if you know that the solution is $f(x) = \sin x$, would you declare that it is explicit? You can barely compute the explicit value of $\sin x$ even if $x = 1$. Moreover, if you allow for a more general class of functions to appear in the expression of $f$, such as $\Gamma$-function or Bessel functions, perhaps you can solve explicitly a bigger class of equations. Finally, if you know that for some integral operator $\mathcal I$ it holds that $$ \mathcal I[f] = 0 \tag{1} $$ has a unique solution $\hat f$, the equation $(1)$ by itself serves as an unambiguous definition of the function $\hat f$. You can see now, that an explicit solvability of $(1)$ is quite a relative notion and it depends, in which class of functions are you looking for the expression of solution.

In the end, to answer your question about the real need for exact solutions, let me mention that it hinges upon the problem in hand.

1. Suppose, that the solution of $(1)$ describes some real phenomenon by itself. Then often numerical solution with precise bounds on the error is enough. Yet again, even $\sin 1$ can be computed in reality only with some imperfect precision.

2. However, if the solution of $(1)$ enters as a part of some problem, it is much more handy if you can express all the parts of this problem using the same language. Suppose, $f$ in $(1)$ further describes some dynamics $$ \dot x = f(x). \tag{2} $$ Even a small perturbation in value of $f$ can dramatically change the character of the asymptotic behaviour of $x$. Not to say, that the purely numerical expression of $f$ makes it hard to do the qualitative analysis of the ODE $(2)$.

I hope, it helps - though I am not sure whether this is the best place to ask such questions, as it may be not considered as a research-level question. Better try MSE