You actually have gotten a bad explanation. There's no such thing as being "a little too fast" which would cause this effect, and there definitely is no "spiraling out" due to inherent speed/momentum.

An object in orbit of another remains in orbit as long as its horizontal velocity is high enough to not be pulled into a collision with the parent, but low enough to not escape the gravitational pull altogether. The closer to the parent, the stronger gravity affects the object, so you have to go faster horizontally to keep "missing" the parent, making gravity only pull you into a circle around it instead. That's why it's also called orbital speed: the object is not going straight in a line, it travels at speed in an orbit.

If you want to change an orbit, you need to accelerate or decelerate. This energy has to come from somewhere. And obviously, the direction you accelerate in matters. If you speed up horizontally, increasing your orbital speed, you'll get further away from the parent, but by moving further away, your orbital speed will decrease and be lowest at your furthest point. Then, if you don't keep accelerating, you'll start to get closer to the parent again, which makes you go faster. This is an elliptic orbit.

If you go fast enough horizontally, you eventually can get so far away that the parent's gravity influence becomes negligible, and the gravitational influence of another parent matters more. This is called reaching escape velocity. If you leave earth orbit, this is usually the sun.

If you were to simply slow down the object in its orbital speed, the object would get closer to its parent until it collides.

If instead of accelerating the object "forward"/horizontal to human observer on earth, you'd accelerate "up"/away from the earth, you interestingly would not cause the object to get further away from its parent. Yes, you'd move higher up, but that would also mean that you equally slow down along the "forward" axis. So as explained before, if you stop accelerating, the object will start being pulled by gravity again until it reaches its now even closer than before proximity to its parent, half an orbit later on the other side. Because it's now closer to the parent, it has sped up and will then start moving away again, another elliptic orbit has been achieved.

And if you accelerate "sideways", so neither away from the surface nor forward along the orbital path, you actually change very little: you only affect the inclination of the orbit. Usually we think of objects going around the equator, but they don't have to. An orbit can go any which angle, even rotating around the poles, going South to North or vice versa.

So long story short, how does the moon speed up? It doesn't have and rocket engines or similar. The reason is the vast difference the earth and the moon rotate around themselves. The earth takes 24h to rotate. The moon takes roughly 27.3 days to rotate a single time. This causes the Earth to "push" the global tidal waves around its oceans much faster than the Moon gets pushed. This actually causes the moon to get "dragged along" a tiny little bit on every tidal rotation. This not only speeds up the rotation of the moon itself: the moon is so slow that it doesn't have time to transfer all that rotational energy before the tidal wave on Earth has moved on the surface to be a bit on front of the Moon. This is the moment where the Earth's center of gravity is a tiny bit "forward" of the middle of the Earth. This in turn pulls the moon forward along its orbital path, speeding it up horizontally. Obviously, this also means that Earth's rotation gets actually slowed down by the same amount.

All these effects are incredibly tiny! The moon moves "away" at 3.8 cm per year, whereas it will take 50 years for an earth day to be a single millisecond longer.