I’m not sure if there is a definitive answer to why this works better, but to provide some insight, its worth noting that using the log-likelihood is very common in statistics. Here are some references on the use of the log-likelihood [1] 280, [2] 89, [3] 74.
One key point to notice is, depending on your loss function, this fundamentally changes the calculation. Let’s consider a case were your true class is 1 and your model estimates the probability of the true class is .9. If you loss function is the L1 Loss function, the value of the loss function is 0.1. On the other hand, if you are using the log-likelihood then the value of the loss function is 0.105 (assuming natural log).
On the other hand, if your estimated probability is 0.3 and you are using the likelihood function the value of your loss function is 0.7. If you are using the log-likelihood function the value of your loss function is 1.20.
Now if we consider these two cases, using the standard likelihood function (akin to softmax), the error increases by a factor of 7 (.7/.1) between those two examples. Using the log-likelihood function (akin to log-softmax) the error increases by a factor of ~11 (1.20/.105). The point is, even though logsoftmax and softmax are monotonic, their effect on the relative values of the loss function changes. Using the log-softmax will punish bigger mistakes in likelihood space higher.