In general, solid crystals are formed when there is either too much of a substance dissolved in a liquid (for example, too much salt in water), or when the liquid is removed and the solid stays behind.

For example, salt crystals stay behind when boiling off the water from a pot full of ocean water. In your case, the crystals will form on a string that is immersed in a solution of water, which also contains a lot of sugar (almost as much or even a little more sugar than would usually dissolve in water).

So why do these neat looking crystals form as opposed to just some solid block of sugar? This has to do with the structure of these substances on a very small level.

The tiny particles that make up these substances (for example, the atoms and molecules that make up sugar or salt) tend to arrange in a certain pattern. This pattern can be a cube, a rectangle, a cylinder, or many other much more complicated structures. This is on a very very small level ... so small that you sometimes cannot even see it with a regular microscope.

So when the first solid crystals form, they will arrange themselves in that manner, which can be different for each substance.

Once more crystals form, they will continue to arrange themselves in that same shape, until we can eventually see the neat looking shapes with the bare eye. You have probably noticed that this can be a little tricky. Sometimes, it works well and you get nice crystals and sometimes it doesn't work so well.

Crystal formation can be tricky like that, and it depends on a number of factors (how much of each substance is there, the temperature, the cleanliness of the containers, ...). Also, on another note, the reason why you need to put the string into the solution is to give the crystals a point at which they can start to form, almost like a "seed".