Let those numbers be $$2n−1$$ and $$2n+1$$. Then the equation
holds true, as you can check by expanding the terms.
Which means that 1 is an integer linear combination of the two consecutive numbers, hence their highest common factor is $$1$$: a number that divides $$2n−1$$ and $$2n+1$$ obviously divides the left hand side of the equation. Then it also divides the right hand side of the equation, which is $$1.$$