You will be able to use fullScreenMode when you want your stack resized or scaled to take full advantage of the available screen that its been deployed too.
What this means is that, you create a stack at any size that you want, and then LiveCode automatically adjusts this to a given screen.
The main reason for the resolution independence feature is so you do not need to worry about the various sizes/dimensions of every mobile device that is available. This is especially useful when it comes to Android-
http://en.wikipedia.org/wiki/Comparison_of_Android_devices
An example of this in action is, when you deploy and non-retina iPhone sized stack (320 x 480 (3:2 aspect ratio) to a Galaxy S3 (1280 x 720) (16:9 aspect ratio).
As the aspect ratios do not match, then using “exactFit” will cause some elements to appear stretched/skewed, “letterBox” will add small black bars to top and bottom of screen, but aspect ratio will be kept in tact, “noBorder” will crop the stack and maintain aspect ratio however as cropping occurs some elements may be slightly cut and finally “noScale” centres the stack on screen with no scaling occurring.
The following is a great site for calculating the aspect ratios of a given resolution-
http://andrew.hedges.name/experiments/aspect_ratio/
and we also have a tutorial which explains how to use implement resolution independence here-
http://lessons.runrev.com/s/lessons/m/15262/l/156477-how-do-i-make-my-app-scale-to-fit-the-screen-on-all-devices
There are also the LiveCode release notes which explains the Resolution Independence features in more detail-
http://downloads.livecode.com/livecode/6_5_0/LiveCodeNotes-6_5_0.pdf
http://downloads.livecode.com/livecode/6_5_1/LiveCodeNotes-6_5_1.pdf
http://downloads.livecode.com/livecode/6_6_0/LiveCodeNotes-6_6_0_dp_1.pdf