You can get all kinds of scientific (and pseudo-scientific) explanations from people, which usually involved something called 'Circle of Confusion' (CoC), at which point the discussion generally breaks down into something involving a religious argument.
However, in general, and from observation, you are right - all things being equal, to the normal eye on a normal sized print, DoF is made smaller by a larger sensor.
DoF is affected by these things:
1) Focal Length.
2) Distance to Subject.
3) Aperture.
4) Sensor or Film size.
For a practical demonstration, if you take a macro photo with your digicam, you'll see that it exhibits thin DoF; that is, things in the background are blurred out pretty well. This is because you've decreased DoF by decreasing the distance to your subject, without changing anything else.
You can also decrease DoF (to increase those out-of-focus effects on the foreground or background) by opening up your aperture (lower f-number), using a longer zoom (more focal length) -OR- increasing your sensor size.
If you want to play around with cause-and-effect, try DoF Master's free online calculator for fun.
http://www.dofmaster.com/dofjs.html
All you have to do is manipulate the four things I mentioned above - and not the resulting near limit, far limit, and total DoF.
Why are these things good to know? Well, from my point of view, understanding and intentionally controlling DoF to the extent that one is able given circumstances, is another way to add creative control to your photographs. Just like experimenting with focus, shutter speed, and aperture, this is something you can add to your bag of tricks.
The most common example of this is in portraiture, when it is generally desirable to have the background blown out into unfocused fuzz to some extent. A common digicam on a sunny day just can't do that - it is limited to deeper DoF due entirely to the small sensor. So a dSLR makes a better choice in that case - a film camera such as a 35mm, medium format, or even large format possibly even more so.
We work with what we have, but it helps to know how to control the tools we have to the extent they can be controlled.