The simple answer (and one that you’ve probably heard before) is that we only see one side of the moon because the moon rotates around the Earth at the exact same speed as it rotates around its own axis, so that the same side of the moon is constantly facing the surface of the earth. This means that one full ‘day’ of the moon (meaning the length of time it takes for the moon to rotate around itself once) is about 4 weeks long. If the moon didn’t rotate at all, we would see all of its sides; the only way for us to see such a constant face of the moon is if it’s also rotating. There’s a great visualization of this below.
If you watch the way the moon spins (or doesn’t), you can see that only the left side has a consistent side facing the surface of our planet (which, we must note, is not even a little bit to scale here).
However, the underlying reason why the moon rotates at this exact speed, forcing us to only see a single side of it, is because the moon has been tidally locked to the earth. Tidal locking is a stable configuration, and relatively easy to get to, given enough time, so many of our solar system’s moons are found to be tidally locked, including the dwarf planet Pluto and its largest moon Charon, which are both tidally locked to each other.
The “lock” part of this name refers to the way that an object - like the Moon - is apparently fixed in position, with one side always facing the other object. Any object which is found to be tidally locked will always have one side of itself facing the surface of the planet it’s orbiting. The amount of time it takes to orbit around the planet will vary from object to object (Phobos, one of the moons of Mars, is tidally locked and orbits Mars every 8 hours - way faster than our Moon), but as long as the object is tidally locked, the rotation will match the length of time it takes to orbit.
However, it’s the “tidal” part of the tidal locking that gives us the real key to why tidal locking happens at all.
We’re most familiar with tides as the effect of our oceans rising and falling due to the position of the moon. The Moon’s gravity pulls on the earth, and the water on the surface of the Earth closest to the moon responds to that pull by elongating towards the moon. The water on other parts of the earth feels the Moon’s gravitational pull as weaker, with the water on the opposite side of the earth feeling the weakest pull. However, these tidal forces also have another effect - they resist rotation.
The Moon was almost certainly not tidally locked when it first formed - at that time, it would have rotated at a faster speed, which meant that had any observer been on the early Earth, they could have seen all sides of the moon as it spun. However, the gravitational pull from the Earth - which like the tides due to the Moon, pulls on the side of the Moon closest to the earth more than the far side, resisted this faster rotation. This resistance due to the gravitational pull of the Earth gradually slowed down the faster spin of the Moon until the Moon was no longer rotating faster than it was orbiting. Once the Moon’s rotation had slowed so much that a single face was always facing the surface of the Earth, it had officially been tidally locked, and has stayed in this configuration ever since.
The Moon also has the same influence on the Earth, but since the Moon is so much less massive than the Earth, this resistance to rotation takes a much longer time to impact the Earth's spin. However, it’s still a measurable effect! The Moon is slowing down the rotation of the Earth by about 15 microseconds every year, gradually lengthening our days.