T cells kill infected cells by detecting foreign proteins using a lock and key tube mechanism (Key = foreign antigen, Lock = T cell receptor). Normally any given T cell an its off spring have only one specificity of lock (although collectively there are many millions of locks in the body). CAR-T therapy is when they pull out a bunch of T cells for a patient and genetically modify the cell to have a know lock which is specific for a key known to be on the Cancer, then put them back into the patient. So, relatively, there is a massive increase in the frequency of T cell with that specificity in the patient, and hopefully they kill the tumor cells.
Cancer = you, but misregulated. Code for various tissues are turned on at 'incorrect' times, including code for profliferateNow(), ignoreAutoDestruct(), and can start running contradictory code like beNeuronLike() and beMuscleLike() simultaneously.
T-Cells generally sense 'rogue' cells, and command them to auto-destruct. Many cancers are rendered invisible to your immune system because their presentation to the immune system is indistinguishable from a healthy cell in that it doesn't present any feature that is not found elsewhere in the body.
In an engineered Car-T cell, an immune cell is reprogrammed by providing the cell with a DNA blueprint for a synthetically engineered protein called a 'Chimeric Antigen Receptor (CAR)'[1]. That engineered protein has two major functions - sense a particular cell type, and activate the T-cell to kill whatever triggers its sensor. The sensing mechanism is relatively arbitrary from an engineering perspective, so we engineer the sensor to sense a particular cell type which has become cancerous. And the attack trigger is a shortcut of the entire immunological activation pathway - skipping straight to sense->attack, bypassing the natural defenses to prevent autoimmune problems.
Because the sensing mechanism of the synthetic protein is modular, these engineered proteins can swap out various 'sensors' to create new therapies relatively easily. Sense any cell that looks like a [lymphocyte/B-cell] and [kill it]. More sophistocated versions of these synthetic proteins can actually start to incorporate and-gates and other formal logic into their sensing capabilities:
if(sense(neuron && muscle)){
killTarget();
}
(where no neuron-like markers should ever show up with muscle-like markers except in a genetically scrambled cell, i.e. cancer).
ELI5: Your immune system fights cancerous cells frequently. There's only a problem when the cancerous cells evolve to avoid this by "hiding" (typically by no longer releasing a "kill me" marker). The therapy basically gives your immune system the tools to identify cancerous cells.