I did a calorie calculator to see how many calories I should consume in a day. I'm getting between 1,300-1,500 in a day. So, on days I work out, like today, I burned around 500-600 calories. Should I cut that off of my 1300 calories and consume more food on those days? Or should I just stay at 1300 without taking off the extra calories I burned this morning?
That really depends on what your goals are. If you're trying to lose weight, then a calorie deficit is a good thing. But if you're trying to put on muscle, you actually want to be gaining weight (albeit muscle mass), so you want to be consuming more calories than you burn.
You said you're trying to lose weight, so I'll elaborate a little more. From what I've read, yes you can consume more calories on days you work out and still lose weight. The reason behind that is the benefits of exercising don't stop at just the calories you burn during the workout. Muscles burn more calories at rest than fat, so if you're doing exercises that build muscle, you can and will need to eat more.
You also get an "after burn" after some exercises. According to this source
This is not to say that you should use exercise as an excuse to eat more than you need to. As you read above, after burn depends on the type and intensity of the exercise you're doing. However, in my experience, calorie restriction is not very sustainable (that is, whatever weight you lose while restricting, you'll gain back if you ever return to eating your normal amount).
Yes, you can eat calories to replace the calories you burned in exercise and still lose weight.
If you are on a calorie restricted diet, the most important thing during the day is the net calories that you eat during that day, which can be defined as calories you consume versus calories you burn. (This is a naive approach, but it works for most people if they can stick to it). When you are losing weight, you want the number to be negative. Based on your current calorie goal, you are probably already on the negative side of that equation just by using the Basal Metabolic Rate calculator - which will have an assumption based on your activity level.
Any calories beyond what you do normally for living are added to the calories you burn so you can keep the net calories the same by eating extra during those days. As Lauren said, you might also have other benefits such as a "after burn" effect, but usually I don't count those, or assume the equipment / calculators that I use to calculate the calorie burn include them.
It is also possible to exercise too much and enter a starvation state, but that usually only happens over many days of very low net caloric intake, less than 1000 calories net.