Take a robot that we want to be able to move from the bottom right corner to the top left corner of a 4x4 matrix full of random holes it should avoid. With holes represented by 1s, it could look something like:
As we want it to get to an exit from a start, we have a natural fitness function: closeness to exit door in smallest number of moves.
The genetic programming approach to solving this is to create random computer programs (the second chapter of your link does a pretty good intro to the tree like nature of this process) and let them loose. The vast majority of these strategies will be/are utterly terrible, things like 'go right once' or 'go left ten times'.
Say we make 100 random programs on our first run. We firstly score them on how well they did according to our fitness function (the random programs that did the best). We take a set % of these to survive and get rid of the rest, lets say 10% survive.
We take these surviving 10% of the best performing programs and use them to create new programs for the next generation by modifying them randomly again, but not completely. Say we randomly modify half their structure and leave the other half as is across however many we want for the next generation. We now let this generation loose again, and again rank, score, take the top 10% and breed a new generation from them and so on for n number of generations.
In this case, if we say left the grid as is, the program would generally come up with a rule roughly like 'go left x4, go up x4) as it solves this problem in the easiest way, but if we were to say, continuously randomise the position of 1s in the grid during this evolutionary process, we will force the program to come up with much more generalisable rules, such as checking the cells it can move into for 1s and not moving into any space containing a 1 etc.
Thus we can build a program with a flexible strategy able to cope with different environments for our robot in terms of number/configuration of holes - much more useful than having to program it for every configuration.
Just like with regular evolution, over millions of trials of taking the top performers and modifying them slightly, these programs become very specialised and high performing, able to solve highly complex games, paths with highly complex features etc.