The secret trick is to keep all lightness information in L.
If you have an RGB picture and you use curves to adjust the values for R (red), you inevitably make parts of the picture lighter or darker, and you will probably have to adjust G (green) and B (blue) as well to get the lightness of the original back. In LAB you have no such problem.
The L value (channel) goes from 0 to 100 and indicates Lightness:
This is more or less a black and white version of the image.The a channel is the axis green-magenta. "A" is admittedly not a good abbreviation for green magenta, but that's the way things are.
Negative values, from -128, indicate green while positive values, up to 128, indicate magenta. In the image above, the dark areas are negative, so they will turn out greenish. The light areas are positive, so they will be more red. Note that the shadow on the flower is almost entirely invisible. A shadow is lightness, not colour, so it should not be visible in the a-channel, and it is not.The b channel is the axis blue-yellow.

Negative values, from -128, indicate blue and positive values, up to 128, are yellow.
If we add the three channels we get this picture:

A typical modification in the lab colour space is to emphasize the colours using curves, like this for the b cannel:

The values that seem to be cut off at the edges here are unlikely to actually be used in the picture. The extreme values of Lab describe colours that the human eye cannot see or that do not exist. The modification to the curve is basically just to make it steeper.
Below is a stone that at first may seem mostly grey, but looking a little closer one can see that parts of it have a shift towards red. With LAB colour adjustment, one can emphasise the red, so a casual onlooker will see it much quicker. At the same time the moss got a little more intense green, and the image becomes more interesting.
Before LAB colour adjustment.
After LAB colour adjustment.Clearly, this is not the only way to increase the colours of a picture. Often, it is quicker to simply increase the saturation. But LAB adds a different kind of flexibility, which may be just what you need.
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