This is not about damping and some such but rather solving an initial value problem with elements that by themselves individually do not satisfy that. Here the initial value problem is given by the requirement that it be causal, ie., that all its solutions must be $0$ for $t<0$.
As an example, connect an inductor and a resistor in series and drive them with an ideal voltage source, an ideal battery whose internal resistance is zero and its voltage between its terminals is a constant $V_0$ by turning it "On" at the instant $t=0$ to the circuit whose initial current at the at instant is $i=0$. Defining the Heaviside $1(t)$ as $1(t)=0$ if $t<0$ and $1(t)=1$ if $t>0$ you have the differential equation for the current:
$$\mathcal D i =L\frac{di}{dt}+Ri=v_0(t)=V_o \rm 1(t)$$
How to solve this? You know that the differentiating $e^{pt}$ you get $pe^{pt}$ and therefore $\mathcal D e^{pt} = Lpe^{pt}+Re^{pt} = (pL+R)e^{pt}$.
If the source voltage were a multiple of $e^{pt}$ say, $v_o(t)=He^{pt}$ for some constant in time $H$ then you would immediately see that the differential equation is solved by the substitution $i(t)=\frac{H}{pL+R}e^{pt}$ whose amplitude is independent of $t$ but with dependence on $p$.
Of course, the current $i(t)$ is a real number, so in this context we should have $\Im p=0$ although any complex $p$ is also a formal solution and because of linearity so is a formal solution formed by any linear combination thereof using various other $p$ values as long as $pL+R \ne 0$.
So if we could formally write $1(t) = \sum_k H_k e^{p_kt}$ for some $p_k$ and $H_k$ then we could also say that $i_k=\sum_k \frac{H_k}{p_kL+R}e^{p_kt}$.
In fact a simple analysis shows that
$$1(t)=\frac{1}{\mathfrak j 2\pi}\int_{\mathcal C} \frac{1}{z}e^{zt}dz$$
where the contour of integration is now on the complex plane $z=x+\mathfrak j y$ for some arbitrary but positive fixed $x>0$ starting at $y=-\infty$ and running to $y=+\infty$ for any $t$ positive or negative.
In other words, as long as $\Re p >0$ we can represent $1(t)$ as the continuous sum of exponentials all in the right half plane.
Therefore the solution to our problem is
$$i(t)=\frac{1}{\mathfrak j 2\pi}\int_{-\mathfrak j \infty +0}^{\mathfrak j \infty +0} \frac{V_0}{p}\frac{1}{pL+R}e^{pt}dp$$
Above the $+0$ just means that we are running our integral parallel with the $\Im p$ axis in the RHP.
How to calculate this complex contour integral using Cauchy's residue formulas and some such you have to study the mathematics of Laplace transformation whose inverse transform is just this integral.