How do the Immune System Cells become synsitized?

does the T or B cells have anything to do with it becoming sensitized

Exposure to a previously unknown antigen leads to antigen presenting cells (cells that process the antigen) to migrate to lymph nodes where selection of B cells occurs - a cell with the appropriate level of 'attraction' to the antigen is selected. This cell then undergoes a process called Clonal Expansion, whereby a population of cells are created to produce the same antibody (immunoglobulin).

When the body is undergoes repeat exposure to the antigen, there is therefore a ready population of cells that can respond in significantly less time (hence, the immune system has been sensitized to that particular antigen).

Hope that helps!