DMAP and imidazole both displace the Cl in TBSCl to form a more reactive reagent. Typically if imidazole is used, DMAP is not required, but adding DMAP may reduce the equivalence of imidazole required. 

DMAP is a well known catalyst for acylation and alkylation reactions like described above, but very often people use without the real need.

For example, the protection of primary alcohol goes pretty well without DMAP, and its presence (or absence) will make no difference. However, in the case of acylation of the tertiary alcohols, DMAP catalytic effect will be very much noticeable - you will see the increased yield and shorter reaction time.

Although DMAP is normally used in catalytic amounts, there is still a need to have at least one full equivalent of base like Et3N in the reaction mixture to neutralize the acid released in the process.

DMAP used as Base to catalyse the reaction and to abstract the proton of alcohol meanwhile the presence of imidazole that used as neucleophile that react with tert-Butyldimethylsilyl chloride reagent to form more reactive intermediate that react with OH group easily