Overall in a nucleophilic substitution, a nucleophile (Nu) becomes bonded to a carbon atom and a leaving group (LG) is displaced.  In bonding terms, this means we must MAKE a Nu-C bond and break a C-LG bond.

Consider the stepwise version of the reaction (an S_N1 reaction), for example the reaction of t-BuCl with HO^-.  Before we can make a new bond to the central C atom, we must first break the C-Cl bond by removing the electrons from the bond and giving them to the more electronegative Cl atom.  This is just an extension of the polar character of the C-Cl covalent bond.
Why can't we make the new bond first ? 

Since we have taken electrrons AWAY from C it becomes a +ve carbocation and since we GIVE electrons to Cl it becomes the -ve chloride ion.

Why doesn't the Cl have a -2 charge since it gained 2 electrons ?  

Do the charges balance ? Yes, we have a neutral molecule going to  +ve and -ve whose charges cancel.

In the second step, the electron rich nucleophile donates the electrons to form a new C-O bond.  Note the curly arrow starts from where the electrons are, the lone pairs on the oxygen atom, and goes from electron rich to electron poor:

Do the charges balance ? Yes, we have a +ve ion reacting with a -ve ion to give a neutral molecule.

Now let's consider the single step version of the reaction (an S_N2 reaction), for example the reaction of CH₃Cl with HO^-, where the bond making and breaking occur simultaneously.
Below we can see the oxygen nucleophile donating electrons to form a new C-O bond and the C-Cl bond breaking by removing the electrons from the C-Cl bond and giving them to the more electronegative Cl. By making the bond changes simultaneously we avoid violating the octet rule at the central C.  Notice how the curly arrows "flow"