Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply two large numbers $a$ and $b$, you found their logarithms, added the logarithms, $z = \log(a) + \log(b)$, and then looked up $\exp(z)$ on another table.
Now, characteristic functions do a similar thing for probability distributions. Suppose $X$ has a distribution $f$ and $Y$ has a distribution $g$, and $X$ and $Y$ are independent. Then the distribution of $X+Y$ is the convolution of $f$ and $g$, $f * g$.
Now the characteristic function is an analogy of the "logarithm table trick" for convolution, since if $\phi_f$ is the characteristic function of $f$, then the following relation holds:
$$
\phi_f \phi_g = \phi_{f * g}
$$
Furthermore, also like in the case of logarithms,it is easy to find the inverse of the characteristic function: given $\phi_h$ where $h$ is an unknown density, we can obtain $h$ by the inverse Fourier transform of $\phi_h$.
The characteristic function converts convolution to multiplication for density functions the same way that logarithms convert multiplication into addition for numbers. Both transformations convert a relatively complicated operation into a relatively simple one.