It's all because of your shutter speed.
The shutter speed is how fast the shutter opens and closes, the slower it is, the more blur you will get. This is even more apparant when the focal length increases. The reason you have no problems with flash or outdoor shooting is because theres enough light getting to the camera - so your shutter speed will be higher.
That picture was shot at 800 iso, 55 mm, 1/15 second shutter speed. Theres your problem, 1/15 is too slow. You want at least 1/(focal length) to get rid of shake from your hands, and then maybe even further still if your subject is moving faster. So for example at 55mm you want 1/60 odd, and at 300mm you'll be wanting closer to 1/300. The built in shake-reduction will let you go a bit slower than this rule, by how much depends on how steady you are with your hands.
Either grab a faster lens, or up the ISO untill you get a fast enough shutter speed to get rid of the blur.
Try Tv (shutter priority) mode with a fast shutter speed next time, but just be aware the higher the ISO, the more noise you will get.
A tripod could help, but not that much. The 55-300 isn't all that heavy, so you should be fine hand holding it. If you are having issues, just try and brace your arms and have the camera right up against your face to minimize any shake, or brace yourself against a wall/whatever is around.