Toggle
You can toggle it off with (spacemacs/toggle-vi-tilde-fringe-off), you can get the list of all toggles in helm-spacemacs accessible via SPC f e h or SPC h SPC in more recent versions. Every toggle in this list have 3 functions associated to them, if we take vi-tilde-fringe then the functions are:
spacemacs/toggle-vi-tilde-fringe (toggle the state on and off)
spacemacs/toggle-vi-tilde-fringe-on (force the toggle state to on)
spacemacs/toggle-vi-tilde-fringe-off (force the toggle state to off)
To get the name of the function associated to a key binding use SPC h d k (or C-h k) and type the key binding, with the vi-tilde-fringe example this is SPC h d k SPC T ~ (in most recent version) and you should get the function name.
Excluded packages
Another way to remove the tildes is to exclude the package vi-tilde-fringe from your configuration by adding vi-tilde-fringe to your dotfile variable dotspacemacs-excluded-packages. This will have the effect of completely removing the package and all its associated configuration from your Spacemacs.
Once again to get the package name you can try helm-spacemacs and type tilde in the helm buffer.