At first (depending on the cancer) the few extra cells might still perform their intended function, but eventually there are way too many cells and they form a solid mass (tumor) that doesn’t perform whatever function the original cells were supposed to perform.
Eventually, the tumor gets big enough that it interferes with the normal functioning of the cells around it. If it gets big enough and is in a bad enough area (say the inside of a vital organ like the liver or pancreas) it can stop the organ from doing its job, and the patient dies.
Usually though, cancer patients die from a metastasis, which is when cancer cells break off from the main tumor into the blood and move to a new part of the body, where they start reproducing all over again and form a new tumor.
This is why catching cancer early (before it metastasizes) is the most important factor in defeating it. If there is only one tumor and it’s in a place that can be accessed by a surgeon, it can be removed. If there are lots of tumors, odds are some of them won’t be surgically removable.
There is still hope if your cancer does metastasize though, because there are many new drugs that focus on teaching the immune system to kill the cancer.
Remember the surface proteins up in the first paragraph? Some cancers express certain surface proteins that aren’t usually found in the body, and several clinical trials have successfully trained the immune system to recognize those proteins and attack the cancerous cells.