Hoppy UK:
bigger sensor = higher magnification? I would thing that it is the other way round isn't it? Smaller the sensor, bigger magnification required to get the picture. Or is it me missing the point?

Anyway, this is why I want to upgrade to DSLR, my bridge camera is amazing but lacks big sensor. This results in lack of shallow DOF and high ISO noise.
Haha yes! But no.
The image projected by the lens is bigger. When you roll out all the calculations (circle of confusion size etc) the amount of sharpness lost by enlarging more from a smaller sensor to get a given print size, is outweighed by the gains from starting out with an image that is smaller to start with.
It is basically f/number x crop factor. So if you start with full frame as a standard (24x36mm) then Canon crop factor is 1.6x (15x22mm). F/number x1.6 is a difference of about 1.3 stops. Compact sensors are relatively tiny. They vary but 4x is a typical crop factor relative to full frame, about 4 stops.
In practical terms, if you compose your your subject with a full frame DSLR at f/11, to get the same depth of field and composition with a crop sensor Canon, you would need to set f/6.9 (and use 1.6x shorter focal length lens to maintain the same composition). Picking up your compact, assuming it's 4x crop factor sake of argument, you would need to set f/2.8.
TBH these numbers don't hold
exactly true when you are comparing extremes like going from full frame to a compact, and there are other variables like the shape of the sensor, the aspect ratio, which is 3:2 for DLRS and 4:3 for most compacts (it's the diagonal measurement that's important). However, given that depth of field is, in practise, a very inexact science anyway because it depends on various factors that don't usually apply a lot of the time (such as print size, viewing distance, what is deemed acceptably sharp and what isn't etc etc) it's a pretty good approximation.