This is a problem involving the Pythagorean Theorem, which tells you how the sides of a right triangle relate to each other. Specifically, if a right triangle has legs of length $a$ and $b$ and a hypotenuse of length $c$, then the Pythagorean Theorem tells us:

$$a^2 + b^2 = c^2$$

We can use the above formula in this problem. If you look at the attached picture, I have drawn the triangle in question. We know one side has length 12, we will write this as $a = 12$. We are told that the hypotenuse has length 6 meters longer than the other leg. If we let $b = x$ represent the missing side length, then we can write the hypotenuse as $c = 6 + x$. Plugging this into the Pythagorean Theorem, we get:

$$12^2 + x^2 = (6 + x)^2$$

Now we need to solve for $x$. We do so with a little algebra. Expanding the square on the right side we get:

$$12^2 + x^2 = x^2 + 12x + 36$$

Cancelling out the $x^2$ and rearranging, we get:

$$12x = 12^2 - 36$$

With a little trick, we write $36 = 12\times3$ and we can write:

$$x = \frac{12^2 - 3\times 12}{12} = 12 - 3 = 9$$

Therefore, the unknown leg has side length $\boxed{b = 9}$ and the hypotenuse has length $\boxed{c = 9 + 6 = 15}$.