When something dissolves, it does not vanish. Its particles spread through the solvent.
In salt water, water molecules separate and surround the ions from the salt crystal.
A solution cannot always dissolve unlimited amounts of material. Beyond a certain point, extra solute remains undissolved.
If the water later evaporates, the salt can appear again as crystals.
Dissolution depends on interactions between solvent and solute. Water dissolves many ionic and polar substances well, but many non-polar substances such as oils much less readily.
A useful way to understand this phenomenon is to separate the driving force from the visible result. Temperature, pressure, concentration, surface area and the nature of the materials can each change the rate or the final state. In real systems, several of these factors often act at the same time.
That is why the same phenomenon can look different in a laboratory, in the kitchen or outdoors. The underlying chemistry or physics stays the same, but the conditions change the balance.