How actually an integral transformation(Fourier and Laplace) works?
I can only explain why they work mathematically, b/c the way time-domain functions get turned into s-domain functions is mathematically-based.

Fourier and Laplace Transforms are based on a mathematical concept known as an integral kernel. An integral kernel is defined to be the \$f(x,t)\$ term in an integral of the form:
$$
F(x) = \int_{t_0}^{t_1}f(x,t)g(t)dt
$$

\$f(x,t)\$ defines a family of functions parameterized by x, such as \$e^{-jxt}\$, for example. There are an infinite number of functions that take this form one for each (uncountably!) infinite number of values of \$x\$ from, for instance, \$-\infty\$ to \$\infty\$.

Let's assume we're given a value of \$x\$, call it \$x_0\$, and a function \$g(t)\$ that we want to analyze. Let's assume the domain is \$\pm\infty\$. The above integral can be written as:
$$
A = \int_{-\infty}^{\infty}f(x_0,t)g(t)dt
$$

The above integral is known as an inner product, analogous to the concept for vectors in linear algebra. Informally, an inner product produces a number \$A\$ which measures "how similar in shape is \$f(t)\$ to \$g(t)\$?".

It turns out by summing every single function of the family \$f(x,t)\$, each with a constant multiplier \$A\$ obtained by evaluating the inner product with \$g(t)\$ at each \$x\$, one can get the original \$g(t)\$ as a result.

Now, to answer your question, Fourier Transform decomposes a function \$g(t)\$ into a set of complex exponentials \$f(x,t)\$ of the form \$e^{-jxt}\$. These are very closely related to sinusoids of an angular velocity \$x_0\$ by taking the magnitude of the resulting complex number.

Laplace transforms are similar, but of the form \$e^{-xt}\$, where \$x\$ is a complex number, and \$t\$ is limited from 0 to \$\infty\$ (\$x\$ is normally called \$s\$ in this transform. I am using \$x\$ for consistency). They don't have as nice an intuitive interpretation that I'm aware of. Wikipedia says the Laplace Transform decomposes functions into moments. Analyzing the values of the Laplace transform for values of \$x\$ of the form \$jx_0\$ gets you back to the sinusoidal interpretation as above.

As Lorenzo also points out, there are convenient properties of integral transforms that appear in the frequency domain, such as convolution of two functions \$f(x_0, t)\$ and \$g(t)\$ become a multiplication in the s-domain.