what exactly does scale mean in scale-space theory?
There's an image processing algorithm called Retinex that uses scales to perform local contrast enhancement. Here's an OK presentation on the Multi-Scale Retinex algorithm for more detail.

The image size doesn't change because you are performing convolution of a Guassian kernel with the image. In practice this is generally accomplished in the frequency domain and the image is padded to avoid the effects of circular convolution. This is standard stuff in image processing where you convolve a 'kernel' against an image. It's a just averaging surround pixels with a 2-D Gaussian function instead of something like a block of ones.

I've seen the choice of sigma be a somewhat adhoc "This looks good to me" selection process. There are ranges published in a few papers that seem to work well for most images. Essentially small values preserve the detail in the image, medium values provide a mix of general detail and global energy, and large values provide more global representations of the image energy. I think something like small is 1-25, medium 25-75, and large is +75. When you convolve and sum all of the scales, with a bit more light math, you get a local contrast enhanced image. The results are pretty good. You can try it out on some images to see the result of each to get a better feel for it.