Check out this animation (for reference, this is material which accompanies the beautiful "Clayden - Organic Chemistry" book).
As you can see, there is indeed an intermediate step (the "transition state"), when the nucleophile and the leaving group are in a intermediate (and unstable, being highly energetic) state (nor attached or detached). Look closely at the geometry of the carbon and the hydrogens: that's the sp2 hybridization of the transition state carbon.
The mechanism (also shown on the page linked) involves an electron pair transfer from the nucleophile (here, bromide ion) to the carbon and eventually to the leaving group (chlorine).
The curved arrow formalism, of course, is just a representation of what goes on in the real scenario: as you can guess from the animation, the charged bromide transfers part of its electron density to the carbon (which carries a partial positive charge!). The chlorine atom withdraws the electron density as it is transferred from the bromide to the carbon, and eventually, when a carbon-bromine bond is formed (full orbital overlap between Br and C), the chlorine atom has gained a full electron pair, which allows it to leave as a chloride ion.
This is the how.
As for the why, it's a matter of energy. The reaction conditions are energetic enough to push the system towards the unstable transition state, and the equilibrium is shifted towards the products because the energetic state of the products is lower than that of the reactants (that is: the products are more stable than the reactants)