How do we get NaClO3 in the reaction between Cl2 and NaOH?
I find that when I'm faced when problems having to do with why the products are formed I make lewis structures. 

if you make a lewis diagram of the $$\ce{NaClO3}$$

First off, you will see that a single Cl will be bonded to three oxygen molecules whom all carry full valance shells which were striped from the hydroxide ions in the reactants. (Remember that in solution the Na and OH will be around as free spectators.)

From the structure you will see the reason why there was a formation of the three waters. Since there were three oxygen stripped from the hydroxides and then recombination with more OH to make a reaction like this: 

$$\ce{OH- + H+ -&gt; H2O}$$

Also notice that on the products side for every molecule of NaClO3 made there will be a molecule for NaCl made. From there you can speculate that when the Cl-Cl bond breaks three oxygen atoms and a sodium atom will surround one of the Cl while another is tied to a Na molecule simultaneously.