Let x = 6+y. We can replace 6+y with x getting the equation |x| = 2.
The equation |x| = 2 has two solutions. The expression |x| represents the distance from x to 0 on the number line.
The two solutions to |x| = 2 are x = 2 and x = -2.
Going from x = 2 to 0 is a distance of 2 units, so is the distance from 0 to x = -2.
The solutions in terms of x are then used to find the solutions in terms of y
For example, if x = 2, then
x = 6+y
2 = 6+y
2-6 = y
y = -4
The other y solution is handled in a similar fashion.
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Side note: The 2 at the end of the original equation does not determine how many solutions there are. We could easily have |6+y| = 3 and still have two solutions.