Consecutive numbers are numbers that are (each) one more than the last number. Let's start with two numbers and make this really clear in the equation:
15 = (x) + (x+1)
I've put in the parentheses so you can imagine that they contain the two numbers. When you solved it, you got x = 7. So that means the other term is (x+1) = (7+1) = 8. Now let's extend the idea to three:
15 = (y) + (y+1) + (y+2)
Each number is one more than the last - do you see how that is represented in the equation above? When you solve this, you get:
15 = 3y + 3
5 = y + 1
4 = y
So the first number is 4, the next is 4+1 = 5 and the last is 4+2 = 6. Try this for five numbers:
15 = (z) + (z+1) + (z+2) + (z+3) + (z+4)
These kinds of things are called arithmetic series. That's where you add up a sequence of numbers each of which is separated by the same difference. In your example, the common difference is 1 (subtract each term from the one following it and you get 1). In the future, you'll study more complicated arithmetic series, but no matter how complicated they look the just boil down to four things: the sum, the first number in the sequence, the difference between the terms in the sequence, and the number of terms you are adding.
Hope this helps!