Since you already know that the repertoire of T cell receptors is developed in the thymus, the missing half of the story in the context of antigen-presenting cells is that MHC complexes do not discriminate between self, foreign and mutated self peptides. The take home point is that selection in the thymus is meant to produce a repertoire of T cells which can overcome this fact.
In all healthy tissues, nucleated cells express class I MHC molecules loaded with self-peptides. A T cell probably isn't going to respond to it, however, because trafficking of T cells into tissues requires signals such as cytokines and chemokines. The T cells circulate to and from the lymph node T cell zones.
In the context of inflammation and cellular injury, self proteins may be released into the extracellular space where they're regularly cleared by macrophages via endocytosis. The extrinsic antigen presentation pathway loads peptides onto class II MHC molecules. In this very context, T cells are trafficked into distressed tissues by activated endothelium, where they interact with tissue APCs. Autoreactive clones that escaped thymic selection have the opportunity, then, to react to self antigens, ignoring regulatory or suppressor cells (ref).
Not just self-peptides, however, but also potential self-antigen mimics that take well meaning T cells directed against bacterial peptides, which happen to closely resemble a self antigen, and pit them against your own tissues (example). Unlike autoimmune T cells that have escaped selection and attack your cells, this invites T cells that have gone through the proper channels to do so as well.
See Cellular and Molecular Immunology, 8th ed. for an even better overview of T cell-based autoimmunity.