How does the sizes of nozzle diameter and the Z-axis layer resolution work? I have a 3D Printer with 0.4&nbsp;mm (400 micron) nozzle length and the printer guide suggests that it can print up to a layer thickness of 60 micron and has a accuracy up to 100 micron. 

What I'm trying to understand is that, how does the 0.4&nbsp;mm nozzle output a filament size lesser than it? How does that process work?  Does it reduce the speed at which the filament flows or something?
Yes, they do reduce the speed. Basically, the speed at which you feed filament determines how large of a volume comes out, and the distance you move the extruder means this volume gets distributed over this distance. However, this is not the whole story.

The size of your nozzle determines only two things: the maximum layer height and the minimum feature size. It does not affect minimum layer height (and obviously doesn't affect maximum feature size) although there are some practical constraints (you wouldn't want to print 0.1&nbsp;mm layers with a 5&nbsp;mm nozzle).

The nozzle squishes the filament down as it is extruded; the filament is basically forced into the gap between the nozzle and the previous layer. There's no limit to how thin of a layer you can print (the 60 micron figure is just something the marketing department made up, you can likely print even thinner layers). So long as you can position your Z-axis with sufficient accuracy (which is usually no problem) you can print arbitrarily thin layers.

On the other hand, the nozzle size does determine the maximum layer height. Your need the filament to be squished a bit, or else it won't stick to the previous layer properly. You can't print layers thicker than your nozzle size, and it is generally recommended to print with a layer size that is at most 80&nbsp;% of your nozzle size (e.g. with a 0.4&nbsp;mm nozzle you shouldn't attempt to print layers thicker than 0.32&nbsp;mm) but this is just a guideline.

Your extrusion width is the width of the line deposited by your nozzle. This is generally (due to the mentioned squishing) a bit larger than your nozzle size. With a 0.4&nbsp;mm nozzle you should set your extrusion width to something slightly larger, like 0.5&nbsp;mm. It is technically possible to print with an extrusion width that is the same as (or even smaller) than your nozzle size, but this results in very weak prints: as mentioned before, you want the plastic to get squished slightly.

They claim an accuracy up to 100 micron. This doesn't mean that you can print features as small as 100 micron (since you can't, due to the extrusion width being much larger). What this means, is for example, that if you print a 10&nbsp;mm cube, you should expect its real size to be between 9.9&nbsp;mm and 10.1&nbsp;mm. Such a cube does not have any features that are smaller than your minimum feature size, but its outer walls can be positioned with greater precision than this minimum feature size.

I should caution you that the 100 micron figure is "up to" and in practice you will have a hard time achieving this.