The database may be saved to a backup file in an older MBD and restored in MBD-2016 using the Database tab.
However, that step may not be necessary at all. Since you have MBD-2016 linked to BLS-2016, any league created or restarted by BLS-2016 will become populated in the MBD.
On your old computer, do a backup of your 2015 league, and save or copy that backup to external storage such as a flash drive. Properly eject that drive, then insert it in your new laptop and launch BLS-2016.
Under the Maintenance menu, choose "Restore League(s)". You will have to navigate to access where the backup is stored on your external storage. Highlight that backup file, then click "Open" at the bottom. On the window that opens, click the "Select" button.
The restore process should complete without interruption since the league is not on your new laptop in any form (as far as I can tell by your post). However, if you get a dialog that says something like "The file '....' " already exists. Overwrite this file?", click on "ALL" (otherwise you will have to click "Yes" for every individual file, and the league itself consists of multiple small files.
Once the restore is complete, you can then open this league (there will be some updating occurring as BLS-2016 has slight modifications from a BLS-2015 structure). Once open, you will be where you last used your 2015 league (expected to be at the end of last season).
As illustrated in the BLS-2016 manual (accessible from the HELP menu, starting on page 196), the TASKS menu is where this is located. Choose "Restart League for Next Season" and follow the instructions presented along the way...most should be fairly self-explanatory, such as selecting a start date for the new season, indicating if there's a minimum number of games bowled for a final average to stand, and so on. This utility will allow you to keep the final averages from last year as starting averages for this year, among other things. The one thing I recommend when you do the restart, when prompted for a file name, to append something like "2016" so that you know you are working with the 2016 file, and not confused with old data. Other than that, go through it and if you still need help, ask away here.
Next season, when restarting change the "2016" suffix with "2017", and so on.
Also for next season: If you install BLS-2017 on the same laptop as BLS-2016, the installer will ask to copy many elements to use with BLS-2017's season. At least, I expect BLS-2017 to do just that, as has every previous BLS going back as far as I've used them (2003). It will also copy over all BLS-2016 leagues during the installation, so you don't have to do a BLS-2016 backup...you will have a copy of the 2016 league in the BLS-2017 Leagues folder, and you would just open that and restart for the next season...and so on.
Always check, download, and install the most recent update of your software to ensure that issues may have been corrected.