Phagocytosis will break down the pathogens but it saves little bits of them (that are harmless) to show off to the T cells. When the T cells see them, they recognise there's a problem and they go and provide their own signal to B cells, activating them.
B cells will produce antibodies after activation, which go and bind the pathogen. This can help in removal in several ways. It can bind and physically block an important receptor that the pathogen uses. It can coat the pathogen and make it easier for the rest of the immune system to recognize it. It can help activate other immune cells.
That was the concise version, would you like more detail or is it clearer?
I know this post is a bit late but I just want to clarify if my understanding is correct:
When say a macrophage identifies a non-self "foreign" antigen, it binds to it and engulfs it. Lysosomes then release digestive enzymes that break down this foreign antigen; however, some antigen fragments are presented by the antigen-MHC 2 markers to Helper T Cells. Once the helper T cells dock into the MHC-2 marker, the helper T cell is activated. This results in Helper T cells activating specific B-cells which produce specific antibodies, depending upon the detected non-self antigen. This is a humoral response, because the antibodies and macrophages that are also activated during the humoral response, attach to antigens travelling within the extracellular fluid and make it easier for these non-self antigens to be engulfed by phagocytes, macrophages, eosinophils, neutrophils, etc. A cell-mediated response is also initiated as Helper-T cells activate cytotoxic T-cells; these cytotoxic T cells carry out their functions by releasing perforin (a protein) which destroys viral-infected cells (i.e. viruses are obligate intracellular pathogens that can only reproduce and grow when they are within the cells of an organism). Hence, this immune response is called the adaptive, "cell-mediated" response.
Memory B-cells are also produced; if a particular infection is caused by the same antigen that has been detected before and the primary antibody response has produced specific antibodies against the non-self antigen, the memory B-cells would have retained memory of the antigen and thus in the secondary antibody response, more antibodies are produced in a shorter period of time, building up immunity of the organism to the non-self antigen, and allowing the non-self antigen to be destroyed very quickly by the immune system. How do memory T-cells differ?