So to start off: I (an electrical engineer who had to take classes on that stuff) regularly get sad when I read the packaging of Wifi equipment, because the marketing speech on that is so wrong it hurts.
For example, you have a picture of a wifi access point with 4 antennas. I have a similar one. package says "4×4 MIMO with high gain antennas". Nope. None of these four antennas is highly directive, has high gain, individually. You can get directivity or gain by combining them in very particular ways, but it's not a property of the antenna as is.
We'll see how that happens!
Let me first address a few misconceptions, or maybe just imprecise choice of words, to make discussion afterwords a bit easier, otherwise my explanation might make no sense, as it would contradict the wording of your question.
I understand that high gain directional antennas are able to make sense of weaker signals because they are more sensitive having multiple elements.
nope! That's a fallacy; having more elements is not the key to being able to pick up more. Think about this: assume I have say, 10000 dipole antennas, connected to a coax cable each, and finally merged through splitters. I just throw them on a giant pile.
That's incredibly many elements, but still on expectation (this being very random) a shoddy antenna!
When using a high gain directional antenna […] they're reflected of a surface in a narrow beamwidth.
Not all high-gain antennas have reflectors! Satellite dishes do, but there's very many high-gain antenna designs that don't. Gain just says "there's a direction in which this antenna emits more power (if used as a transmit antenna) than the average of all direction"; high-gain is a bit of a vague term, but basically means there's a direction where you see most power being emitted, which "suggests" it gives you a beam.
transmitted waves are able to travel further
Ah, that one I'll have to veto. Waves travel infinitely long, until something absorbs them. It's however clear what you meant: The signal strength is stronger at a larger distance. And that's just the effect of having gain: if I put in some power $P$ and it's emitted in all directions evenly (so, gain is 1 = 0dBi, "decibel compared to an isotropic antenna") by my antenna, then it doesn't matter in which direction my receiver (of some effective surface area $A$) is: the amount of power it catches only depends on the distance, not the direction. If I, instead, have gain, then less power goes in my direction if I'm not standing in the "main" direction, but more if I stand there.
Important message here is, by the way, that gain just "redistributes" power from all to one direction. No free lunch, an antenna that has high gain means that one direction gets more power, at the expense of the other directions, which receive less.
Now, that's all very "high-level" looks at antennas. You were wondering about a low-level fact of antennas. And, sadly, for that, this abstraction level does not suffice. We'll need to understand how electromagnetic (EM) waves actually work, what they actually are.
So. You know capacitors, I'll assume:
simulate this circuit – Schematic created using CircuitLab
Symbol is pretty descriptive of a plate capacitor: you apply a voltage across the two terminals, you get a potential difference between the left and the right plate. In between, there's an electric field, which goes from the positive to the negative potential, it has an end and a start:
Homogenic electric field. Author: Phatency on creative commons
Change the voltage, you change the electric field. Invert the voltage, invert the field.
You also know electromagnets: You apply current flowing through a coil of wire, you get a magnetic field. That magnetic field makes nice, closed lines going one time around the coil and closes in the coil. That works with 100 turns of wire, it works with 2 turns of wire. It also works with a single wire, in which case the magnetic field simply circles around the current:
Drawing of magnetic field around a current carrying wire. Author: Stannered on wikimedia commons.
You change the current, you change the magnetic field. You invert the direction of the current, you invert the direction of the circling. (You might have heard of the right-hand-rule.)
Now comes the interesting part that has to do with antennas:
Physics has fixed some relations between the changes of magnetic fields and the electric field, and changes in the electric field and the magnetic field. These are Maxwell's Equations.
These equations say that if you change an electric field, you cause an electrical field that looks as if it could have opposed exactly the current that caused the change in magnetic field.
When you change the electric field, you cause a magnetic field, that looks as if it was caused by a current that causes the change in electric field.
Huh. So changing one causes the other; the way these things are linked over distance limits the speed of change, so that not every change has an immediate effect everywhere.
So, and here's how antennas actually work, when you let a current flow through a conductor, and change it periodically, like a sine, then it causes a magnetic field, which also changes sine-like. Which in turn causes a sine-like changing electric field. Uff, complicated, but it turns out the maxima of the magnitude of the magnetic field have a 90° time shift to these of the electric field.
Now, these changing fields have effects on the neighboring places, which now also change sine-like. Congratulations, without actually writing down a formula, we have gotten an idea why a small piece of wire with sinusoidal current through it has field energy propagating away from it, in all directions. It's an antenna! The "thing", the combination of ever-changing electrical and magnetic field, radiating away? That's the electromagnetic wave!
Now, let's talk about gain. All these fields, and their derivatives (what I called "change"), are linear, that means if you have two capacitors and put one between the plates of another, the field in between the inner one's plates is the sum of the fields that the two capacitors would have in itself, for example.
Now, you put two such elemental antennas next to each other, then depending on how you feed them with current (in phase, anti-phased, something in between) and how far they are apart, you get points where the fields of the waves always add up to become twice as strong, and others, where they always add up to zero and cancel. We call that interference. And it's how you can build antennas that have gain. Make it so that parts of the antenna system emit waves with just the right delay so that in one direction you always get the constructive interference, i.e. stronger amplitudes.
You can do that with many individually driven antennas (like the 4 in your Wifi access point!) or you can have one antenna and a big reflector with a very particular shape (like a satellite dish) so that there's a direction where the fields always add up constructively. You can even start by calculating a shape for the conductor so that the conductor through which you let your current flow has gain of its own (like a patch antenna!).
Here's an illustration of these individual (round) wavefronts add up in the arrow-marked direction if the phases between the individually driven antennas are chosen appropiately:
Animation showing how a phased array works.
That's about how your wifi access point "points" a beam in the direction you have your laptop: it choses the right delay for the signals going to its antennas such that the sum of all waves adds up positively in the direction of your laptop. Or some other far-away station.
So, back to: Why are these things reciprocal?
It's simply because if you change the magnetic field around a coil, or a single wire, it induces a current, just working the opposite as if you caused the magnetic field change by changing the current. (That's why motors work as generators, and vice versa). And because an electric field with a conductor running through it causes a current in that conductor.
So, the underlying field/electrical potential/current relationships that you see in everyday circuits have an inverse, and that's why the apply to antennas, too. That's why antennas are always reciprocal (unless they're made of something that is a very strange material, which you basically never encounter, unless you're in the business of building powerful radar systems, or light-amplifying crystals).