What exactly is Laplace transform? I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out that it gives "less familiar" frequency view.
My question is how does Laplace Transform give frequency view? 
I don't understand the connection between $f(t)$ and $\mathscr{L} (f(t))$. For example:- let $f(t) = t$, $\mathscr{L}(t)={1 \over s^2}$   
$f(t) $ gives time view but how does $1 \over s^2$ give the frequency view? Somebody help me to understand what exactly is it. Thank you!!
Can anyone explain it in some physical phenomenon? Like harmonic oscillator?
$$ \ddot {x} + \omega_n x = f(t)$$

Each point in the s-plane is to be evaluated in the Laplace Transform Eq,
i.e, $$H(s) = \int_0^\infty h(t)e^{-st}dt$$
Since $t$ goes from $0 \to \infty$, then $e^{-st}$ can be thought of as an infinite sequence.
Therefore the value $s$ of $e^{-st}$ provides the equation defining this sequence, and the integral of $h(t)$ to this sequence generates a measure of h(t) to this sequence.  
What I still dont understand is e.g, if $h(t) = \exp(+kt)$, then when $s = +k$, i.e, 
$$H(s=+k) = \int_{0}^\infty e^{+kt}e^{-st} dt = \infty$$ important?
Yes, it will give the pole, but so will all other $s$ that are outside convergence?
