To see Bamboo build results in Stash

1. Set up an application link from Bamboo to Stash that uses Basic Auth for outgoing authentication. See Configuring Basic HTTP authentication for an application link.
2. Set up a build notification on the Bamboo plan. This can only be of the All Builds Completed type. See Configuring notifications for a plan and its jobs. If using plan branches, you need to select the "Use the plan's notification settings" from either under the main plan's branches tab or for each branch separately, to receive notifications for branches, otherwise it will use the default settings: "Notify committers and people who have favorited the created branch". The notification options for each branch will be removed as stated in  BAM-14198 - Getting issue details... STATUS

See also Bamboo integration in the Stash documentation.