How do I make video tutorials of someone(s) working through a website on their computer?
Camtasia is probably the leading software for creating video tutorials, it has the ability to smooth mouse movements, the editor gives you the ability to add text, transitions and other effects as well.

The software is quite expensive. It has a free trial for 30 days, however videos exported will have watermarks. 

If you already have access to video editing software, you may be able to reproduce the same effect setting keyframes for start and end positions of a mouse cursor graphic and animating between those two points. You would need use screen recording software that hides the cursor as you will be using your own graphic of a cursor.