Before working out an algebraic proof of the statement maybe try viewing it another way.
Remember that |z1 - z2| = distance between z1 and z2.
|z + 1| = |z - (-1)| = the distance between z and -1.
|z - 1| = the distance between z and 1.
So the statement that |z + 1| > |z - 1| is equivalent to the statement that z is closer to 1 than -1. Think on why this is equivalent to Re(z) > 0.
Picture the points in the complex plane that are closer to 1 than -1. Picture those points that are equidistant.
I hope this helps you work out the proof yourself, but if you'd be interested in some more, please let me know. I'll be happy to explain further. Thank you!