You will need to know how to add inequalities to isolate variables like you do when solving simultaneous linear equations.

x + y + z > 0 then z > - x - y (Let's call it inequality [1])

(1) z > x + y +1 (let's call it inequality [2])
Add inequality [1] with inequality [2] yields
2z > 1
z > 1/2
You can verify this answer by choosing z = 3/4, x = 0, y = -1/2 and z = 2, x = 0, y = -1/2.
Insufficient.

(2) x + y + 1 < 0
You can only add inequalities if the inequality signs are the same. Multiply -1 on both sides yields:
-(x + y + 1) > 0
-x - y - 1 > 0

Add the inequality above with x + y + z > 0. You get:
-1 + z > 0
z > 1
Sufficient.

The answer is B.

Courtesy: GMATQuantCoach