Consider the fallacy: POST HOC ERGO PROPTER HOC (FALSE CAUSATION).
There are several reasons why we should doubt the causal relation between concealed gun permits in Chicago and the metrics in this image.
First of all, Chicago allowed people to apply for permits only at the beginning of the year; it takes some time to actually be granted the permit. There were also several restrictions placed on gun carry to make the law change practically insignificant. Guns were still prohibited in schools, buses, trains, bars, government buildings, and the like.
Next, note the date on the meme. This is not a full year’s worth of data. Does this matter? Of course, seasonality! Chicago has less crime in the first few months of the year. This is because the first quarter is the coldest, and people spend less time outside. Crime rates for the first half of any year are expected to be lower than those of the entire previous year due to seasonality. This could explain the “percent decrease” stats. (Also, we do not know whether a 20% decrease is statistically significant, or within normal variation year to year).
Finally, violent crime rates have been declining for decades across America as a whole. The murder rate in 2013 was 4.5, making it the lowest rate since at least 1960. This makes the “56 year low homicide rate” for Chicago less impressive.