To get the update just reboot your Raspberry Pi.
Raspbmc’s June update is small, but sweet. Here’s what’s new:
XBMC 13.1 final with a ton of fixes for some of the bugs introduced in Gotham:
Fix resume point of certain PVR items
Fix issues with UPNP
Fix crash after AirPlay finishes
Fixes for FLAC tracks with embedded images
Fixes for M4A and MP4 playback
Fix a minor memory leak
Use a unique client broadcast ID in the PVR API
Some fixes for improving support using external soundcards (HifiBerry and iqAudio), see this thread here
Reverted to ext4 as the default filesystem for Raspbmc. ext4 provides better data redundancy and is more compatible with Linux systems for those wishing to backup or transfer data.
Fix issue where XBMC will keep telling user a new version is available.
Fixes for some issues with 3D content