TITLE: What exactly is Laplace transform?
QUESTION [72 upvotes]: 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?
$$ \ddot {x} + \omega_n x = f(t)$$

REPLY [3 votes]: Its easy when you question yourself how is time and frequency are related. You remember frequency is the inverse of time. So if frequency is $a$ then time is $\frac{1}{a}$. Now let us say you have a continuous set of time values that is, a continuous function representing "time" values. Lets say a high tech camera recorded numbers of meteors in a region in sky.Let us say that function is $y=t$, that is, for each sec(any appropriate time unit) same number of meteors were found in that region. For example at $10$ secs $10$ meteors were found and so on. 
Now what is frequency? It is the inverse of time. So in our example we would say how many meteors were found in that region per seconds.But remember previously we plotted time vs number of meteors. This graph is fixed.Forget about that graph. We want a new graph that tells us what is the frequency corresponding to time values. We would like to plot time vs frequency. $a$ and $\frac{1}{a}$ graph. So we take each $x$ points as values of time and each $y$ point as values of frequency. What you asked can be plotted in one graph and understood easily.
Now you see values of time on $x$ axis and values of frequency on $y$ axis. Take any value on time domain and find its $y$ values. You get corresponding frequency value. For $\text{time } = 1$ , $ \text{ frequency } = 1$. As you can see when time approaches $0$ frequency approaches infinite. Don't ask where are the meteors in this graph, that was just for explaining what time values actually are. You can latter plot frequency vs number of meteors.