Completing a prior subpanel install
First thing I would do it get a larger sub-panel. Those dinky 6/12 panels are terrible for space and future expansion. You also need a means of disconnect at the outbuilding. I would get a 20 space main breaker panel which would correct both of these issues. 

In the sub-panel you would also need to install a grounding bar to the back of the panel box and NOT install the bonding screw. The green wire would terminate there as well as the required grounding electrode (ground rod typically) conductor.

At the house you MUST protect that URD cable where it is exposed. Typically we'd use Sch80 PVC conduit. For the splice you can/should use Polaris connectors. They are available at any electrical supply house or even some home centers. They make larger aluminum wire splices a breeze!

There are a LOT of other code requirements for a job like this. Are you experienced enough to tackle this job?