If water is essentially incompressible, why are there tides?

incompressible: not able to be compressed.

compressible: In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change.

So water does not change its volume. The same volume of water can take many shapes.

When no other gravitational force except the earth's $1/r$ potential acts on the oceans, the equipotential surface is defined by the single potential, and the oceans settle in the shape of the equipotential. Because the earth is not really a sphere, these equipotential surfaces vary, but the idea is the same.

When an opposing $1/r$ potential, as the moon potential, is strong enough at time $t$ to add a $-1/r'$ ($r$ from center of the earth, $r'$ from center of moon) potential, then the equipotential form into which the water will settle is disturbed at that time of closest approach.

This is what the solution of the problem gives:

Figure 2: The Moon's gravity differential field at the surface of the Earth is known (along with another and weaker differential effect due to the Sun) as the Tide Generating Force. This is the primary mechanism driving tidal action, explaining two tidal equipotential bulges, and accounting for two high tides per day. In this figure, the Earth is the central blue circle while the Moon is far off to the right. The outward direction of the arrows on the right and left indicates that where the Moon is overhead (or at the nadir) its perturbing force opposes that between the earth and ocean.

So it is the ability of water to change shapes that generates bulges, and the motion of the moon that changes in time moves these bulges with respect to the earth.

Please note that there exists also earth tides, i.e. the ground also bulges as far as the elasticity of the solids it is composed of allows it. Also note that water tides can appear differently in different locations due to the geology of the ocean bottom and the land boundaries, and just because water is a fluid and obeys equations of fluid flow.

I think you have started from a picture of the tides making water go up and down that is a natural picture to have, but it is wrong: they don't. If that picture were right, you ought to be able to measure the tides just by digging a deep well.

What happens is that the influence of the moon (and sun) changes the shape of what is meant by "level" - technically, the equipotential surface, the surface where the gravitational potential energy is equal across the planet. Anna V's picture is the one to look at here.

Locally, this change in global shape translates into a tilt. A piece of sea or lake which was level a few hours ago now feels that it is not level any more, because the equipotential surface has moved on (following the moon). So its water does what any water does in those circumstances, which is flow sideways from "high" to "low" until it is level under the new definition. And water doesn't have to be compressible to flow sideways.

A good model of this is to take a big flat shallow tray and half fill it with water. Then tilt it from side to side. The surface of the water will end up horizontal (because that is the equipotential surface in your experiment), but it will do it not by stretching up and down but by sloshing from one side to the other. (The difference here is that the tray is moving relative to the equipotential surface, rather than the equipotential surface moving relative to the Earth, but the effect is the same).

The sloshing of the water in the tray as you move it is also important for tides. The difference in height between high tide and low tide in a perfectly frictionless perfectly inertialess ocean is only 18 inches or so. The fact that we get far huger tides in real life is down to the sloshing of real water in sea beds, and round coastlines of interesting shapes.