The "log unit" term refers to a unit increment change in that variable, converted into the logarithmic equivalent.
The Table 3 on p. 158 shows how the variables that the researchers investigated are related to the output variable, which is ELO rating.
The variables included:
v1: Hours of serious study
v2: Hours of tournament play
v3: Years of private instruction
v4: Years of group instruction
v5: Current hours/week serious study
v6: Current hours/week tournament play
Each of these is converted into a logarithm before the equation's factors are calculated. (The factor of a variable is the multiplier of that variable that produces the closest prediction to actual results, when all factors and variables are included.)
Their regression model shows that (if you use only these variables and the logarithmic approach) the peak ELO of a player will be best predicted by the equation:
PeakELO = 1145 +
195 x log(v1) + 32.7 x log(v2) +
9.4 x log(v3) + 4.3 x log(v4) +
6.3 x log(v5) + 20.3 x log(v6)
where 1145 is just a constant that helps improve the prediction.
They point out that the variables with the strongest effect on the prediction were those with the highest factors; v1 (total hours of serious study), and v2 (total hours of tournament play), but the first was much stronger than the second (195 vs 32.7, which is nearly 6 times as strong a factor), and dominated the other factors in their effect on the prediction.
The short answer to your question is that a log unit for the variable of total hours of serious study is an increment of 1, which transforms into different numbers of actual hours, depending on what the increment is being added to.
To illustrate what this means, if I've studied 1,000 total hours, then the log of 1000 = 3. My predicted PeakELO (for that variable alone) becomes
PeakELO = 1145 + 195 x log(1000) = 1145 + 195 x 3 = 1145 + 585 = 1730 ELO
(I'm ignoring the other variables for simplicity, but the results show that they're not substantial anyway.)
If I work 1 more hour, then
PeakELO = 1145 + 195 x log(1001) = 1145 + 195 x 3.0004 = 1145 + 585.086
which is an indistinguishably small change.
If I work 100 more hours, though:
PeakELO = 1145 + 195 x log(1100) = 1145 + 195 x 3.041 = 1145 + 593 = 1738 ELO
So, while I'm at 1730, 100 hrs more of study will boost my ELO by 8 points.
But look at what happens if I've only studied 500 hours in total so far:
PeakELO = 1145 + 195 x log(500) = 1145 + 195 x 2.70 = 1145 + 526 = 1671.
Now, if I study an additional 100 hrs,
PeakELO = 1145 + 195 x log(600) = 1145 + 195 x 2.78 = 1145 + 542 = 1687.
The same 100 hours boosts my ELO by 16 points.
What the formula is clearly doing is reflecting the diminishing returns of the same number of hours of study time as your ELO increases. It accomplishes this accurate depiction of what actually happens by using the logarithm of the underlying data.
However, I would question the simple interpretation that the investigators offer, which is that a log unit increment of 1 results in 200 ELO increase. Well, log of 1,000 is 3. Log of 10,000 hrs is 4.
So, they claim that I would get 200 ELO improvement if I had already studied 1,000 hours, and now committed to studying another 9,000 hours (yes, that's about 4.5 years of full-time study). In their model, I would advance from 1730 ELO to 1930 ELO in no less than that time.
Clearly, this cannot be true for every player. I would suspect the following:
a) some confounding variables are having an effect, such as effectiveness of study (as determined by periodic assessment), ability to implement knowledge in tournament games (as determined by post-game analysis vs established knowledge base), age of player which the investigators acknowledge the model does not take into account, and so on...
b) the model's coefficient of correlation estimate (R^2) is only 0.34. Perfect correlation has 1.0; chaos (no correlation) is 0. So, 0.34 is promising, but not wholly effective by a long shot. Also, the Standard Estimate of the Error is 222, meaning the predicted ELO could be off by over 200 ELO for a particular player, and it wouldn't be remarkable. That's a pretty wide margin. So, in short, the model needs substantial improvement to be useful as a predictor of ELO.
What the model does accomplish at the moment is to suggest that study is 6 times more effective than tournament play, which is in turn more effective than any other training method, including coaching, for the average player.