When you submerge an object of certain weight $w$ in a fluid (water tank), there is a buoyant force exerted by the fluid to the object $B$ pointing upwards. This means that the upward force on the object exerted by the hand $F$ is smaller for the magnitude of the buoyant force
$$F + B - w = 0 \qquad \rightarrow \qquad F = w - B$$
Third Newton's law of motion teaches us that for every action there is a reaction. That means that if the fluid is pushing the object up with a (buoyant) force of magnitude $B$, the object is also pushing the fluid down with a force of same magnitude. Since the water tank does not move, it follows that the (ground) surface has the push the water tank up with a force
$$N - w' - B = 0 \qquad \rightarrow \qquad N = w' + B$$
where $w'$ is weight of the water tank and $N$ is normal force exerted by the surface. It is exactly magnitude of this normal force that the scale measures.
Why does the weight increase for denser then water objects even if they are still being held?
This has something to do with how buoyancy works. By definition, the buoyant force magnitude equals the weight of the fluid which has the same volume as submerged part of the object.
If object is less dense than the fluid, it will float on the fluid and you do not have to hold it. When object floats on the fluid without you holding it, then $F = 0$ and $N = w + w'$.
If object is more dense than the fluid, it tends to sink and you have to hold it if you do not want the object to sink. In that case the scale would measure $N = w + w' - F$, and if you decide to let go of the object then $F = 0$ and $N = w + w'$.