Howdy. If you read the manager's manual, you will find some answers on how the setup and interface works.
If you want the "guts", the best method is to build one (vicibox) or even just pop in the demo (also on vicibox, demo) and then look through the files and read the comments in them.
call_log in this case isn't a "script". I note that when you asked:
you changed the code slightly from the subject line:
the :4577 is extremely important. There is a listener script running on port 4577. This script has a function for call_log, which is what is being invoked.
I'm not sure if I'm describing it correctly, but it's as close as I can get really. The point is: you cannot ignore the :4577, or ANY individual component or even character in a script or a dial plan (even if you know what it is). This software is unique, and accomplishes a task that was heretofor overwhelming for those attempting to write "dialers" in asterisk. As such, Matt (the creator and maintainer of the software, along with his company) has put a great deal of time into making the system work "Just So".
If you build a system (or fire up a demo) and verify that it works ... then you could try some modifications and see what can/can't be ignored/deleted/revised. Until then, if you want to learn the system ... you have two ways to go: install it as given (do not learn how the internal combusion engine interacts with the fuel recovery system, JUST TURN THE KEY, the engine will run ...) OR get ready for a serious learning curve.
Of course, you could do like I am: BOTH. Use it as it is, learn as you go, and make changes to suit your own needs (and the needs of your clients). If you do that, it works better. Also, if you have a live (functioning, in production) system, the learning curve seems much better.
So if you are curious: Start with finding out what is listening on port 4577 of 127.0.0.1 and what will happen in that script if you invoke "call_log" as a switch/argument/function and whether data will be stored or the "call" modified through an agi call or both or what. It is all quite intricate, but also right there in the code.