If you ever take apart a VCR and look inside (something you will probably want to do on a dead VCR), you can now understand a good bit about what you are seeing. Here is a typical view inside an RCA VCR:
Taking a closer look at the drum, you can see that it is tilted with respect to the tape:
When the VCR loads the tape, the two rollers identified by the green arrows will actually pull the tape out of the cassette. They will move along the tracks identified by the red arrows and wrap the tape around the drum:
Here's a better look at the two rollers:
When the tape is first seated in the VCR, these two rollers are actually inside the cassette, behind the tape. The cassette has cutouts to allow these rollers to fit into the cassette. In this picture, the tape is clear, and you can see the rollers behind the tape:
Once the rollers fully extend on their tracks, the tape fits very nicely around the drum:
The pinch roller and inertia roller engage, and the tape is pressed onto the erase and audio heads. It's like watching a ballet when you see all of this take place -- there are some pretty amazing mechanical engineers working on this stuff!