Metathesis reactions, or double replacement reactions, are reactions in which the products are formed by the exchange of ions present in the reactants. In other words, the cations (positively charged ions) and the anions (negatively charged ions) exchange partners.
Ionic compounds are formed when a positively charged ion and a negatively charged ion come together, the strong electrostatic force being what keeps them bonded to each other.
These compounds form lattice structures and are solid at room temperature (more here: http://socratic.org/questions/why-are-ionic-compounds-solid-at-a-room-temperature).
In order for the cations and anions to exchange partners, which is what is needed for a metathesis reaction to take place, these ions must be separated from each other and from the lattice structure that holds them. Here is where water comes into play.
Solids cannot react until moisture is present. When dissolved in water, the electrostatic attraction between the ions is broken, and ions can move freely in solution.
This is what drives the metathesis reaction - the removal of ions from solution. The most common metathesis reaction involves the formation of an insoluble solid, or a precipitate.
To sum it all up, dissolving ionic compounds in water is what allows for the exchange of cations and anions, which can then move freely in solution and form either a precipitate, a weak or non-electrolyte, or a gas.
Here's a video that provides some examples of metathesis reactions: