Are they trying to blame immigrants (read, Mexicans) because a black man can't keep a job or stay out of prison?
Nope. They just established an econometric relationship. Where every 10%-induced rate of immigration should reduce the black wage of that group by 2.5 percent, lower the employment rate by 5.9 percentage points, and increase incarceration rate by 1.3 percentage points. Though there could be omitted variable bias in the regression, which I doubt, but it would be meaningful to see what potential variables could affect it and re-run the regression. They do mention White male statistics also got worse, but not as sharply. This could largely mean that there are a higher % of Blacks considered low-skilled.
Though I could see it as somewhat true though, immigration would shift the supply curve for low-skilled labor outward reducing the equilibrium wage (putting downward pressure) though it is somewhat sticky heading downward and the decrease in wages would lead some to drop out of the labor force (assuming you're heading downward on the labor supply curve and the substitution effect is taking place) and once you're out you will have more time to be put in jail.
There was another paper famous in Labor Economics authored by Card and Krueger and found that the Mariel Boatlift actually reduced Black unemployment in Miami over the long run, though this was found using the Differences-In-Differences method.
