We know that sqrt(n) is rational if and only if n is a perfect square.
We assume X = sqrt(n-1) + sqrt(n+1) is rational. Hence X^2 is also rational. X^2 = n - 1 + n + 1 - 2sqrt(n^2 - 1). Therefore sqrt(n^2 - 1) must also be rational. But that means n^2 - 1 must be a perfect square. n^2 is already a perfect square, so this is clearly impossible.
You can also show that n > sqrt(n^2 - 1) > n - 1, meaning no sqrt(n^2 - 1) is not an integer, hence n^2 - 1 is not perfect square.