The short answer to your question is no, eating fat does not cause you to store more fat.
What happens to a given food when your body digests it is actually very complex. See for example this and this. So it is hard to say things "eating X causes Y" or "not eating X causes Y". Furthermore, there is a lot of variation among individuals.
The good news, however, is that to a good approximation there are two major things you can do to burn excess fat: 1) eat at a caloric deficit 2) eat less carbs. Beyond that, what you eat won't matter too much for the purposes of weight control. (You should probably worry about getting micronutrients and some other things but that's a separate discussion.)
What constitutes a caloric deficit varies by individual, but it is relatively easy to figure out. First you need to know what your Basal Metabolic Rate is. For example by using this calculator. That's how many calories your body needs to function -- it does not take into account any activities you do. Depending on your level of activity people say to multiply the BMR by 1.2 (not very active) to 1.5 (very active). To be at a caloric deficit you need to eat less calories than that number, BMR*activity-multiplier, each day. 600 calories less per day translates approximately to 1lb of fat per week.
The second thing, eating less carbs, is very simple. Track the food that you eat for a few days and try to have less than 100g of carbohydrates each day. After a few days of this your body's glycogen stores start running low and your body switches to burning fat instead. This is the principle behind all the low-carb diets (keto, atkins, slow-carb, ...).
Experiment with yourself to find out what works. There is significant variability between individuals.