Screw's head screwed up
The easiest way I've found to remove screws which you just cannot get purchase on with a screwdriver is to make the slot bigger. This can be accomplished with Dremel type rotary tool with a slim cutting disk. This is made easier if the head of the screw is above the working surface such as in the picture you showed. All you are trying to do is recreate the slot you've damaged. This will probably mean you'll need new screws in the end (but I think you are there already anyway). This, of course, will not work if the head is snapped off. Once the new slot is created, just use a flat tip screwdriver of the appropriate size to remove the screw. If it is still happening after this, the aforementioned processes of screw extraction would come into play.