There are many different kind of anchors with different working principles (see wiki). Most of them, however, rely on the chain lying on the ground for the anchor to have its effect.
To understand this, let us look at a stockless anchor and how it holds onto the ground (picture from the wikipedia article):
From the picture it is clear that the anchor has its main strength when the chain pulls parallel to the ground. In fact, the way the anchor is liftet is by pulling it upwards (pciture also from the wikipedia article):
So what about the chain stretching/slackness? Let's look at the following picture (from this quora question):
So essentially what happens is that most of the chain lies on the ground. The rest is fairly loose, which gives the ship some swaying room, but when pulled by wind in one direction it will stretch sufficiently to balance the wind force on the boat. The weight of the anchor chain therefore also plays a crucial role, since the chain should not be stretched fully by the wind force.
In summary, if we stretch the chain too much, the anchor cannot operate. Instead, a large part of the chain has to lie on the ground, such that the force on the anchor is rougly parallel to the ground. The weight of the anchor chain ensures that forces on the boat such as waves and wind do not fully stretch the anchor chain.
Aaron Stevens pointed out that the question also asks about the equilibration phase after anchoring, which I missed initially in my answer. Specifically, why one waits until the anchor relaxes again.
Let us look at what happens after throwing the anchor.
- Anchor hits ground.
- Ship still moves. Anchor is dragged along.
- The anchor gets a grip. Remaining movement of the ship stretches the chain, drags the anchor further.
- Eventually, the anchor gets stuck in its final position, with the chain most likely stretched due to the ship movement.
- The ship is stopped and dragged into an equilibrium position by the stretched chains weight.