Why APC's present antigen to T cells?
Let's consider some situations real quick:

(1) A bacterial microbe enters through a fissure in the skin. What does an innate, systemic immune response look like? In short: it's lengthy, uncoordinated and has a lot of potential to damage tissue in the area of infection.

(2) A bacterial microbe enters the blood. The bacterium produces an AB toxin that enters cells and kill them.

(3) The host is infected by a helminth.

In situation 1: Innate immune responses are associated with the secretion of many cytokines and chemokines. Cytokines are associated with the inflammatory response, and chemokines are associated with lymphocyte trafficking. Inflammation and recruitment of lymphocytes, macrophages and the like are good for clearing the infection, but there's a lot of potential to damage the surrounding tissue. Effector cells from the adaptive response incl. humoral immunity don't elicit the same level of damage, and we can illustrate this in mucosal immunity. Since commensal (and some pathogenic) organisms are constantly stimulating mucosal immunity, there are (1) epithelial effector T cells, (2) plasma cells and (3) dendritic cells that make the response to these pathogens fast, efficient and little damage occurs to the tissue involved.

In situation 2: So let's say the pathogen has escaped the immune response, maybe it's encapsulated. There are AB toxins being produced that kill cells endocytosing them. What good will it do for a phagocyte to eat something that ends up killing it? Antibodies are particularly efficient for this extent because they can neutralize toxins they've encountered with the same fast, efficient response as we described above. This is achieved through antigen presentation. The host will then need to find other ways to clear the infection, however.

So: In situation 1 we illustrated that an innate, systemic response isn't always desirable due to tissue damage, and through the presentation to T cells by APCs, a later adaptive immune response mitigates this effect through formation of effector cells. In situation 2, we then illustrated that the first-line defenses can't always respond to the imminent threat for various reasons, which are overcome by the presence of effector cells.

Situation 3: This is the more complex concept. You have different subsets of CD4+ T cells that can be formed in response to the type of interleukin or cytokine that is present when the CD4+ T cells are being activated. This differentiation is driven largely by secretions of activated APCs! The APC performs this critical step based on what kind of receptors the specific types of pathogen are detected by. The differentiated CD4+ T cells, or T-helper cells of class: Th1, Th2, Th17, Treg, etc. each perform different roles which are effective in different situations. For example, you wouldn't see any benefit in throwing Th1 cells at a helminth infection, Th2 is the best effector for this. Treg cells are capable of abrogating autoreactivity, and Th17 cells are needed to assist B cell differentiation in response to antigen detection (which in the case of helminths, the B cell must class switch to IgE and this is stimulated by Th2 cells via IL-4).

Now, in situation 3, we've illustrated that CD4+ T cell intermediates are providing your overall immune response: flexibility, specificity, and checks/balances which are all-at-once critical to healthy functioning. I disregarded MHC-I responses and CD8+ CTL's because this is about APC-driven responses.