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Can someone please explain what approximate solution actually is
An approximate solution is a path that gets closer (in terms of the joint state space) to the goal, but still doesn't reach it. The path is still feasible/not in collision, but it doesn't actually get to the goal, it will just stop somewhere in the middle.
why can't it be executed
Just a guess, but there's no good reason to try to execute an approximate motion. Say I asked a robot to pick up a cup of water. Then it finds an approximate motion that stops while the gripper still 5 feet away from the robot. As a user, I would be very confused what it was doing and why didn't actually pickup the glass.
As far as why you are only finding approximate solutions, this is probably because you are only running RRT* for 5 seconds: depending on the complexity of your environment this might not be enough. Try running for longer (in Rviz there is a parameter you can use to set the planning time).
Is there an option to force moveit to execute the approximate solution?
Given that I found this question by trying to google for this option, I would guess not. :) After actually looking at the source code, in MoveIt!, I can confirm that there is currently no easy to execute an approximate solution.
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