Here's a classic motivating example. Let $H$ be the Heaviside unit step function
$$
H(x) = \begin{cases} 1 & x > 0 \\ 0 & x \leq 0 \end{cases},
$$
and suppose we want to solve the differential equation $$u' = H.$$ We attack each part of $H$ separately:
\begin{align*}
x > 0 &\implies u(x) = x + c_0 &\mbox{ for some } c_0 \in \mathbb R, \\
x \leq 0 &\implies u(x) = c_1 &\mbox{ for some } c_1 \in \mathbb R.
\end{align*}
Now our differential equation has $u'$ in it; this means that $u$ is differentiable, and hence $u$ is continuous. Continuity of $u$ at $0$ enforces the constraint $c_0 = c_1$. Hence, letting $c := c_0 = c_1$, we get
$$
u(x) = \begin{cases} x + c & x > 0 \\ c & x \leq 0 \end{cases}.
$$
We've run into the problem: we cannot differentiate $u$ at $0$, as we get $1$ from above and $0$ from below. Hence there is no differentiable function $u$ satisfying $u' = H$. Uh oh.
Let's see the magic that happens if we turn our differential equation into an integral equation by first multiplying by a "nice" test function $\varphi$ and integrating. By nice, we require that we can take derivatives of $\varphi$, and that $\varphi$ vanishes at infinity. Our integral equation is
$$
\int_{\mathbb R} \varphi u' = \int_{\mathbb R} \varphi H.
$$
Let's see if $u$ works in this equation. Starting with the LHS, and integrating by parts
\begin{align*}
\varphi u \big|_{-\infty}^\infty - \int_{\mathbb R} u \varphi' &= - \int_{\mathbb R} u \varphi ' \\
&= - \int_{-\infty}^0 c \varphi' - \int_0^\infty (x + c) \varphi' \\
&=- \int_0^\infty x \varphi' \\
&=- x \varphi \big|_{0}^\infty + \int_0^\infty \varphi \\
&= \int_0^\infty \varphi \\
&= \int_{\mathbb R} \varphi H,
\end{align*}
where we've integrated by parts again and used our "nice" properties of $\varphi$.
The upshot: $u$ did not work for the differential equation, but it did work for the integral equation. This is the point of using integral equations instead: they allow for greater possibility of the existence of solutions because solutions aren't required to have such stringent regularity conditions (like differentiability and continuity). We "shift" those requirements onto the "nice test function" $\varphi$.
