OK, I feel dumb now.
Here's a great little video that explains how the tides work. I always thought the tides were water being pulled up by the moon's gravity... which on reflection is obviously impossible.
But as the video explains, the right way to conceptualize them is that the moon's gravity is squeezing the oceans in low tide areas, which pushes up the water in high tide areas.
It all boils down to the differential between the moon's gravity effects on the planet as a whole versus various specific items on earth's surface, depending on where they are on the surface relative to the moon. Any particular item on earth's surface will be pulled toward the moon at a slightly different angle, and at a slightly different force, than the planet is pulled. (In other words, a golf ball in China being pulled toward the moon is not being pulled parallel to the direction the planet as a whole is being pulled, unless the moon is directly over the golf ball at the time.) The differential between those vectors creates downward or sideways pressure sufficient to create 'low tide', which in turn squeezes up the water elsewhere to create high tide.
You can tell I'm no scientician.