“Why do they inject dye for a CT scan?”

Contrast (known to lay people as "dye") is not really a dye. It has no color. For CT scans - it contains Iodine. This stops X-rays, and looks white. It reveals where the vessels are, how they look, and also where there is blood flow. So, some tumors with a lot of blood flow will become much more obvious.

Contrast improves the visualization of anatomy, tumors, infection, inflammation, and organ function. It helps evaluate arteries and veins. It dramatically improves the interpretation of the scan.

It lights up abnormalities.

IV contrast (dye) is injected during a CT to help the radiologist make a more accurate diagnosis. Initially, the dye travels through the vein to the heart and lungs. This phase of enhancement is useful in determining the anatomy of the veins and the presence of possible blood clot (pulmonary embolism).
It then re-enters the heart and is pumped to the arteries and rest of the body. During this phase, arterial anatomy can be discerned to check for problems such as leaks (extravasation), narrowing (stenosis), dilatation (aneurysms), or malformations and shunts.
The contrast then enters the tissue and small vessels. The enhancement pattern of organs can help identify and characterize different conditions.

It helps to outline the abnormal areas.