First and foremost, Kungsträdgården is not the tunnelbana station, but a nice square and park in…read morethe center of Stockholm by the water with a view of the palace.
In August, there are usually a few stages with live music as part of the Stockholms Kulturfestival (http://kulturfestivalen.stockholm.se/ ). It's really worth checking out the festival program, if you're in Stockholm at that time. During the year, there are also many other events, so you should not be surprised if you go there and something is going on.
When it's suny, Kungsträdgården is a great place to enjoy the sun. There are many benches around, and also a kind of fountain where many people hang out when the weather is nice. It's not only close to the palace, but also to the opera, Sergels torg, Östermalmstorg and other locations in central Stockholm. That also makes it a good meeting place.