How many times does a certain sound appear in an audio clip?
How can I count these hit sounds ( I should get six in this clip).


In this case you can simply rectify the signal (i.e. run it through an absolute function, then through a simple running average low pass filter and at the output of this add a threshold. Essentially, when the power of the input signal exceeds the threshold you will be receiving a pulse whose width is approximately equal to the duration of the "hit". Alternatively, you can do away with rectification and filtering and simply pass the whole thing via a threshold operator. That is, when the power of the input signal exceeds the threshold, you get a "pulse". In either way, count the "pulses" and you can count the number of "hits" that you have in that wave file. 


  What is the best tool/software/app for this problem.


This is not an incredibly challenging Digital Signal Processing task, so you could simply write a Python script that does it in 3-5 lines of code (I am not implying that Python is not capable of serious processing here). Alternatively, you could install something like scilab and again do it in 3-5 lines of code, using one of its toolboxes (or, this one).


  Where can I find a good source that explains the theory behind this.


This "rectify and threshold" technique is a sort of a "standard" in counting pulses, alternatively, you might want to have a look at the moving average filter as an example of a sliding window technique where you process a long signal in small "frames" of samples. Other than this, this website has an extensive list of book references if you search for "books".

Hope this helps.