Long ago the magnet was far far away and the magnetic flux through the coil was essentially zero.
If you measure the induced emf as a function of time with time as the x axis and the induced emf as the y axis then the area under the curve an above the x-axis between the vertical lines $t=t_1$ and the vertical line $t=t_2$ is the total change in magnetic flux from time $t_1$ to time $t_2$. This is a law of physics, known as Faraday's law. It is not explained in terms of anything more fundamental, and is motivated by its agreement with experimental observations.
Apply that to the previous paragraph and the area under the emf graph up to some line $t=T$ is the actual magnetic flux through the coil at time $t=T$.
So as the bar magnet falls the flux gets larger and larger until the magnetic is literally inside the loop. You can see this by drawing the magnetic field lines around the magnet and imagining different places a loop could be. If centered on the loop all the field lines come through the middle of the magnet and out the north pole and each one goes in a big loop before it turns around and comes back in through the south pole of the magnet. Each one has some distance away where it is fully pointing the opposite direction as when it goes from north to south. Those that turn around outside the coil contribute to the magnetic flux because the one going north contributed, and the same field line was outside the coil when it pointed south. Those that turn around within the loop go in and out and net don't contribute any flux (they contribute an equal amount positive and negative flux). If you pick a field line that pointed south outside of the coil (the ones that contributed on their way north but on their way south) and you imagine placing the coil farther up eventually the coil and the field line intersect if you put it even farther, the field line now fails to pierce the area enclosed by the coil as doesn't contribute any flux at all. So the flux goes down. Each field line had such a point, so the flux just goes down and down and down as the coil is farther from the magnet.
You can do the same thing by sliding the coil downwards but this time the field line crosses the plane of the coil outside the coil on its way to the south pole of the bar magnet. So the net effect is that the flux is weaker the farther the coil is from the magnet, and when the magnet was centered on the coil the flux contained all the field lines that didn't turn around within the coil (it had those but it had them coming and going so no net effect). And it only gets weaker than that.
So when the coil is farther away from the magnet you have less flux in the area enclosed by the coil. So we expect the area under the emf curve to be zero way to the left, and then as you imagine moving a vertical line for $t=T$ to the right you start getting more and more area to the left of the line $t=T$ until the area equals the area magnetic flux when the coil is centered on the magnet. That's when the flux is biggest. At this point the flux has to get smaller as the magnet continues to move. We can make the area smaller by saying the area below the line $y=0$ Volts and above the emf curve counts as negative area. So we need to get that, and eventually when the bar magnet is far away the total flux is zero again so the negative area between the emf curve and the x axis (the area under the x-axis and above the emf curve) needs to equal the positive area between the emf curve and the x axis (the area under the x-emf curve and above the x-axis) so the total area is zero, to have the magnetic flux be zero. As it needs to be at late times, i.e. far to the right.
All we have left to explain is why the area to the right is more squished. Since the magnet is falling, it speeds up and fall faster and faster, so it moves away from the center fast later on so the flux changes faster later on. So we need to accumulate more area more quickly so we need a larger emf, a larger emf gives more area in a certain time, for instance the strip of area between the vertical lines $t=t_1$ and $t=t_2$ contains more area when $t_2-t_1$=1sec (in one second) if the emf is larger (and positive area if the emf is positive and negative area if the emf is negative). So the last part of the curve needs to look like a squished version of the first part, but even more squished nearest to the right.
As far as I can tell this is the first answer that doesn't talk about rates of change, and explains the why of each step in layman's terms.
Technically if you want to know which direction the emf goes you need to pay attention to whether you drop the magnet with the north pole up or the north pole down. No ideas change, the same kind of emf curves happen either way.