Using TBAI (tetra-butylammonium iodide) as a catalyst. Ideas on the mechanism or the driving force?
I was wondering, what is the driving force for the Iodide to bond to the carbonchain instead of the bromide?


It's not so much a driving force as much as it is a dynamic equilibrium of sN2/sN1 reactions. Halides can exchange each other in equilibrium if one is not removed thus processes that remove the unwanted product such as the Finkelstein reaction are use to achieve complete substitution (think Le'Chatlier's Principle).

$$\ce{R-Br +I- &lt;=&gt; R-I + Br-}\qquad\quad\ \ \  \text{(Equillibrium)}$$
$$\ce{R-Br +I- +Na+ &lt;=&gt; R-I + NaBr (s)}\qquad \text{(Finkelstein Reaction)}$$


  Is there a mechanism to explain that?


Given the information you provided, Your reaction is proceeding as follows:

$$\ce{R-Br + I- -&gt; R-I + Br-}\tag{1a}$$

$$\ce{R-I + R'-O- -&gt; R-O-R' +I-}\tag{1b}$$
Where as the reaction:
$$\ce{R-Br + R'-O- -&gt; R-O-R' +Br-}\tag{2}$$

cannot proceed or does not proceed at a fast enough rate and thus the catalyst can improve the reaction rate by first substituting the less reactive halide. The Wikipedia article on Williamson ether synthesis agrees with this:


  However, if an unreactive alkylating agent is used (e.g. an alkyl chloride) then the rate of reaction can be greatly improved by the addition of a catalytic quantity of a soluble iodide salt (which undergoes halide exchange with the chloride to yield a much more reactive iodide, a variant of the Finkelstein reaction).