Hiding markup elements in org-mode

Try:

(setq org-hide-emphasis-markers t)

or set it via customize:

M-xcustomize-variableRETorg-hide-emphasis-markersRET

I like to show/hide these often. It you want to make it a simple toggle, put this in your init.el to get a C-c e binding:

(defun org-toggle-emphasis ()
  "Toggle hiding/showing of org emphasize markers."
  (interactive)
  (if org-hide-emphasis-markers
      (set-variable 'org-hide-emphasis-markers nil)
    (set-variable 'org-hide-emphasis-markers t)))
(define-key org-mode-map (kbd "C-c e") 'org-toggle-emphasis)

In case you're using Spacemacs, you can also toggle this with M-RET T V (toggles space-doc-mode). It hides org-mode emphasis markers and meta tags, among other things. For a full description, see e.g. SPC h f space-doc-mode.