With start.me, you decide if you want to open links in the same tab or in a new one. Click the Account icon in your top right corner and open the Settings.
Locate the toggle option Open links in new window and set it on. It will turn bright green. Now every time you click a bookmark, you'll get a new tab.
If you leave this option untoggled, your bookmarks will open in the same window or tab you're currently viewing.