Hi. First of all: My English is very creative, also I dont always use correct terminology, sorry for that.
You will have to add a custom function in your functions.php. When I created mine (get_event_list), I changed the get_events function that comes with event calendar as default. I didnt now what I did, that means I'm not completely shure if everything is correct. Maybe theres code left that isnt needed.
What the code above does (the WHERE-Thing) is to scan for a given category and only return posts from that category. If you want to do as less as possible, you can just add that part to your copy of get_events. At the top of the function youll have to modify the part whith the variables, so that you can tell the function which category to display. In your template file you can use $cat to query your current category.
If you like, you can try changing the rest of the function, to display your calendar another way. I did that, it was a litt bit of try and error.
I forgot what the 200 is for. I think something like how many posts should be displayed. It is not neccesary for what you are trying to do, just leave it there.
What get_events and the other functions do, is looking in the database for calendar dates and wrapping them in HMTL-Code. When you call the function inside your loop you can tell the function, how exactly the dates should be wrapped (Thats what the link is for). If you need it to display in a very special manner, you will have to modify the function itself. (The ec3 documentation would be useful, but its currently offline).
If you have to know more, you will have to be more specific with your questions, I dont think my english is good enough to write an understandable tutorial for this.