2 possible ways. The first, for lower levels of current, it can disrupt the electrical signals of the heart. This is usually how people die from household current. The heart is regulated by electric current, and an outside current can make the cells contract and not stop contracting, preventing it from beating normally.

The second way, usually for very powerful current, is by destroying internal organs and other body tissues via heat damage. This is how lightning bolts, high power lines, and electric chairs work. That's also how that guy in that one awful "wrecked" video is able to crawl after grabbing the wire inside a high voltage transformer. It cooked half his body, but missed his heart, so it took him a couple of minutes to die. Don't play with electricity, kids.