New light switch causing other switches to not work
While looking at your picture makes it difficult to tell how you wire nutted the wires in for your new switch (those neutrals are in the way - if only they were pulled out so we could follow the wiring back to each connection). That said I will take two stabs at the issue here and the problem is most likely one or the other.

Your Z-Wave should be configured like this:


Line In : Bottom Left.
Neutral : Bottom Right.
Load : Upper Left.
And if it has 3-Way traveler : Upper Right.


The most likely scenario I suspect is:


  Perhaps when you added in that Neutral Line for your Z-Wave you did
  not get all of those neutrals together in the wire nut very well - I
  have known this to happen to people not just on the neutral but the
  hot side as well - the line in side just did not get wrapped up with
  all of the other wires , because the installer simply added one wire and tried > to use the same wire nut, which was fine before but a tad too small now and the 
  wire slipped down a bit and did not make connections with all wires.


Another possibility is:


  Perhaps you disconnected the line in that would be considered the
  power distribution line serving all 3 of those switches and then you
  attached it to the load side of your Z-Wave?