well, dw...that is not quite correct. You are downloading the videos. In order to play, a file must be ON your machine. Usually, unless you take steps to keep it permanently, it is in a temporary folder and will be deleted either when a certain capacity is reached, or when the browser is closed, depending on settings. (You can go TO that folder, rename the video, and move it to a folder where it will be kept, if you wish. They don't make it easy or explain it, but it is not that hard if you want to do it. There are also ways to directly download videos TO the folder you wish.)
(btw...this is the same with images. ANY image you view is kept in a cache or temp folder, often/usually with a coded name that is not relevant to the picture.)

To address Leenia's question, I have not heard of malware in YouTube, but I suppose it is theoretically possible. I would imagine Google takes step to be relatively sure that all the files IT hosts are safe.