In this case, the y variable has a noticeable positive skew. It's hard to tell from the relatively low-resolution figure of the original variable, but it's possible that there is an abrupt cutoff just to the left of the mode.
This might be an indication of a variable that has an artificial left truncation. The "true" value of an employee to a workplace, for example, might range from high positive values to even negative values. If you try to ascertain this from paid wages, though, you will encounter an artificial left cutoff in the form of a minimum wage. Taking a logarithm of this type of variable can make it more normally distributed (as in the case you show). Say you're considering the effect of number of school years and recorded felonies on wages. The true model might be one that includes values that have been artificially been truncated.
This, however, leads to the question of whether taking the logarithm will improve the regression's performance. Saying that an increase in $x$ will cause a linear increase in $y$, is very different than saying that an increase in $x$ will cause a linear increase in $log(y)$ - these are contradictions, in fact.
Linear Regression Models with Logarithmic Transformations describes the four conceptual possibilities of taking logarithms in regression (a logarithm can be taken on a dependent and/or independent variable).
I think the important point is that the decision of whether to take a logarithm of the dependent variable depends not only on it, but also on the dependent variable(s). For one, as the paper above explains, it really depends on what the true model is. Also, note that, to some extent, linear regression assumes that the residual is normally distributed, and the residual depends on transformations to all variables.
So while the graphs you show are some indication that a log transformation on the y might be beneficial (it makes it more normal), you can't fully decide that based on the y alone.