First of all, it is important to determine what kind of substitution reaction the halogenoalkane will undergo. We are given the information that it is a primary halogenoalkane, which generally undergo a SN2 reaction. 'S' standing for substituion, 'N' for nucleophilic, and '2' standing for bimolecular. The reaction is SN2 and not SN1 as the carbon centre is not sterically hindered (the nucleophile is able to 'barge' through), so attack by the nucleophile is possible. Furthermore, the transition state is more stable than the carbo-cation that would otherwise be formed.
In this particular case, the hydroxide ion attacks the carbon atom attached to the bromine in the bromoethane, as it has a partial +'ve charge, forming a high-energy transition state. The the carbon centre being bonded both to the hydroxide ion and halide ion for an intensely short period of time (imagine a weightlifter with two extremely heavy dumbells being carried by both arms). The reaction proceeds with the halide ion 'breaking off' heterolytically, the carbon-hydroxide bond being fully made, and the new subsituted product, ethanol, being formed. The second-step is the rate-determing step, thus a second-order reaction.