You correctly describe that the tide is caused by the gravitational force of the Moon. The magnitude of the gravitational force depends, among other things, on the distances of the bodies. Therefore, gravity acts the most on the water on the near side of the Earth, and therefore, there is high tide. However, we cannot solve the whole situation only with water but we also have to take the Earth into account. The first part (the high tide on the near side) is settled. At a greater distance than the water, is the Earth itself (we will be interested in its centre of gravity because there is the gravitational force). The Earth is also attracted to the Moon. We can think of such an attraction like falling. So water falls to the Moon the most on the near side and the Earth itself falls a little less towards the Moon (because it is further and therefore less attracted). And the water on the far side falls the least. Therefore, the surrounding water flows into this “empty” place and there is high tide on the far side.

The fact that we do not fall towards the Moon is because the attraction of the Moon causes the rotation of the Earth (Moon).This rotation is much more visible on the Moon than on the Earth because the Earth is many times heavier than the Moon.

For this explanation, I used the description from the point of view of space. In a description from the point of view of the Earth, I would have to use the term centrifugal force. The result is the same — the low tide will be on both sides of the Earth because on the near side, the centrifugal force has the same direction as the Moon’s gravity and therefore reduces the gravitational attraction of water by the Earth. On the far side, there is the least gravitational force of the Moon to the water and therefore the centrifugal force will have a bigger effect and the surface will rise.