Fitness helps you develop strength, no doubt about it. You don't need to be able to run marathons, but if you cannot run 5km however slowly without stopping then you will simply not be able to handle heavy lifting to any significant degree.
Put another way, you might get from beginner to intermediate in strength without much fitness (though it will improve a bit even if you don't train it directly), but you'll be stuck at intermediate and never reach advanced if you don't do any fitness stuff.
If you do try to run marathons, you'll hinder your weight training. Your body can only muster so much resources for improvement, you can get both fit and strong, but unless your diet and rest and so on are absolutely spot on you'll find it hard to get very fit and very strong.