Damping is indeed pretty much self explanatory: it's the rate at which vibration energy is dissipated within the system. What it means for the sound is not so simple. The problem is not so much in the guitar end: you can measure the wood and determine it's damping factor (with some reservations) and look at the sound of the assembled guitar to figure out how fast the vibration is being damped out and at what frequencies. The problem is relating that to the player's and listener's perceptions, which are far from simple.
Low damping systems have two things in common: they 'ring' for a long time once they've gotten started, and they have a well defined frequency. A material that has high damping, such as foam plastic or particle board, gives a short 'thud' when you hit it, with no discernible pitch. A piece of aluminum or glass can have a very well defined pitch, and ring on for a long time. Wood in general is between those extremes.
You have to keep in mind that the damping of a particular object will be the sum of all the things that can dissipate energy. This will include material properties, of course, but also other things such as the sound that is radiating from it. Something like a loud speaker may be made of materials that have low damping, but it doesn't weigh much, and has to move a lot of air when it moves, so it won't 'ring' when you tap it. This plays out in wood as well. I used to think that balsa wood had high damping until I got hold of a thick piece to test. It turns out that the wood itself is in the same general class as spruce for damping, but, being low in density a thin piece just hasn't the weight to keep going. This sort of damping is probably most important in the guitar top. Most of the literature I've seen on this says that the 'loss' to the system from sound radiation is 'small' relative to most other sources of damping, so it's hard to say how significant it is.
Wood is a complex material, and there are several ways aside from making sound that could dissipate vibration energy. Some of these could well vary with frequency. We usually measure the damping factor of wood by vibrating a sample at it's resonant pitch, but the exact frequency will depend on the size of the sample and so on. There's no guarantee that the number we get there, which is usually at a pretty low frequency, has any relationship to what's happening at other pitches. It could be that otherwise similar woods could have very different damping at high frequencies, for example, which might bear on the way the guitars sound.
Since damping tends to cut down on high frequency response you might think that a guitar made of low damping wood would have a lot of highs, and sound 'bright'. As it happens, though, Brazilian rosewood and Western red cedar both have notable low damping, and people often say that guitars made of those woods tend to sound 'full' or 'dark'. Maple and spruce, with higher damping, are usually said to make 'bright' guitars. It's difficult to reconcile these statements. It may be that the ear and brain are adding all of the energy in the overtones to the fundamental, giving the instruments with more overtones a 'darker' sound. It's possible, too, that since cedar and rosewood are darker in color, they sound 'darker'. The higher damping woods will tend to give a sound that both starts an stops more quickly, and that 'punch' might come across as 'brighter'. There are a lot of dimensions to this. It's not easy to sort them all out.
If you could make two instruments that were 'the same' except for the damping factor of the wood used it would be fairly simple to say what it meant in terms of sound. That's not easy, and may well be impossible. There has been at least one computer modeling study that looked at damping as one variable, and found that it had little effect. OTOH, the model was very limited as compared with a real guitar, which is an engineering nightmare, so it's hard to say how much weight we can put on that. Another study based on a different electronic setup did find some interesting things about damping. In particular, they noted that either too much or too little damping tended to produce sounds that listeners did not like. This could be one reason we don't particularly like aluminum or plastic guitars. On other hand, that study was on violin sound, and guitars may be somewhat different.
In the end nothing about the guitar is really 'simple'. That's what keeps it interesting (and frustrating!).