Imagine that only the Earth and the International Space Station exist. The ISS is doomed because it has stopped orbiting, and is falling straight down. Inside the ISS, an astronaut has placed 5 marbles in the shape of a plus sign (center, left/right/up/down). For convenience, the "center" marble is also at the center of mass of the whole ISS. There's no air or anything to mess up the results. What happens to the marbles?
Look at the problem from the frame of the Earth. The Earth is exerting a certain force on the center marble, call it F. The "down" marble is slightly closer to the Earth, so it experiences a greater force, F + Something. The "up" marble is slightly further from the Earth, so it experiences a lesser force, F - Blah. The "left" and "right" marbles experience forces very very close to F (slightly less, if they're in a straight line, but much closer to F than F - Blah). However, those forces are directed towards the center of the Earth, so those vectors have small components directed towards the "center" marble.
Now that we've figured out the forces, look at the problem from the frame of the marbles. We're in free fall, so the "center" marble floats motionless. The "down" marble is going to float further away from the center marble (F + Something pulls "down" harder), and the "up" marble will also float further away from the center marble (F - Blah pulls "up" less than everything else). The "left" and "right" marbles float towards the center.
Using only Newton's laws, this reveals that gravity's behavior over a volume has a stretching effect in the up-down direction, and a squeezing effect in the left-right direction. GR confirms this. This effect is the tidal force.
When applied to the Earth and the Moon, this explains the two bulges. The bulge closer to the Moon is due to F + Something, because that water is being pulled stronger. The bulge away from the Moon is due to F - Blah, because that water is being pulled less than everything else.