Step one has you log onto the PSN Store with a PSP and download the UMD Passport App. Step two is to run the app with the UMD you wish to transfer in the drive. The PSP will then connect to the PSN and authorize a UMD Passport download of the game. From there you log into the PSN with your Vita and buy the passport version of the game.
While not all titles are available for transfer, Sony also released a list 50 game companies partaking in the passport program as well as a list of the games that will be transferable when the service has it's first major update on March 9 in Japan.
Sadly, Sony told announced last month that this service would not be coming to North America, so those outside of Japan will just have to pay full price and re-buy digital versions their favorite UMD titles on the PSN.
Update:Originally this article referred to March 9 as the UMD Passport program's launch date instead of what it actually is: a major update. The UMD Passport program actually launched on December 6 of last year in Japan