Can I lose weight and fat entirely through strength training?
Technically, yes, kind of. :/
In reality, probably not.
First, I notice you ask about losing both "weight" and "fat." It's great that you understand these are different, or rather, that fat is a subset of the "weight" you can lose. Most people who want to lose weight actually want to lose fat.
First, if you're doing strength training, you'll be putting on new muscle, which will add weight. This is a good kind of weight to add. Every muscle cell in your body burns calories just to keep itself alive, 24/7, even when you're asleep. Adding muscle to your body will increase the rate at which your body burns calories, all the time, which makes it easier to lose fat and keep it off. If you want to lose fat, you want to add muscle.
Second, for most people, if you start working out without paying any attention to your diet, you'll get hungrier and start eating more. Not a lot, maybe not enough that you notice, but it'll happen. And the result of that is that your calorie consumption will increase until it matches your new calorie expenditure, and you'll stop losing fat.
For most people, the only way to effectively lose fat and keep it off long-term is to tackle both ends of the equation at the same time - both diet and exercise.
Google "eating clean" for a great diet, high in fruits and vegetables and low in fat and protein, that's ridiculously healthy. I'm on Kathy Abascal's diet To Quiet Inflammation, on my doctor's recommendation; it's pretty much the ultimate in eating clean. Abascal has a book on Amazon if you're interested. I've lost about 90lbs on this diet, along with exercising, over the last two years.
While eating a healthy diet, start strength training. I enjoyed heavy lifting until health problems forced me to stop, but you can also do bodyweight exercises (Google that for some great examples - there are videos to show you exactly how to do them) at home if you can't afford a weight set or a gym membership.
Start slow. Beginners who are just starting to exercise tend to be too enthusiastic and often hurt themselves, which sets you back by however many days or weeks it takes to heal, and can completely discourage you. Work on your form, whatever exercise you're doing - make sure you're doing it correctly before you start pushing intensity. If you're lifting weights, that might mean working with only the bar for a while before adding plates. For bodyweight, it might mean going more slowly, or doing fewer reps, while your body becomes accustomed to this new exercise. If you're a little sore the next day, that's fine. If you're a lot sore, throttle back a bit. Be patient - you'll get there!