A soft opening occurs when ride construction is finished, but it has not been officially opened to the public.
The ride will be opened to park guests sporadically and without any announcement. They just open the ride and let people ride until whatever test they are running is done, or the ride breaks, or they are done adjusting whatever they are adjusting.
We rode Test Track several times this way before it opened. We would either be walking by, or hear the ride start up. This would be our que to run over and see if we could get on. Several times we did manage to get a ride.
This allows the park to test rides without people being annoyed that the ride goes "down" for maintainance or adjustment. Once the kinks are worked out, the official opening occurs.