It is fast, or an attempt to be fast. It's also clumsy There is no sense of unwilling and none of tired. They were perhaps surprised in the middle of doing something else and had to stop what they were doing. Or they were really comfortable and cosy. But they tried to get up fast and for some reason did it clumsily.

"Scramble" is a fast, inaccurate movement, usually using all four limbs (arms and legs). For example, scrambled eggs, means eggs that are quickly stirred over heat. The sense is that the effort is difficult, thus fast or crude.

It fits with a description of kids scrambling off a sofa as they may have their feet on the furniture and are, therefore, more like to "scramble off" it.