When 2m+3>0, 2m>-3 so m>-3/2 and 2m+3<13, 2m<10, m<5 so -3/2<m<5.
When 2m+3<0, m<-3/2 and -(2m+3)<13, or 2m+3<-13, 2m<-16, m<-8, so -8<m<-3/2.
Another way of solving this is to write 2m+3<13 and 2m+3<-13 (in other words 2m+3<+13 and solve both equations).
Therefore the solution good vets two ranges for m: m between -3/2 and 5 and m between -8 and -3/2. Note that m cannot be equal to -3/2, 5 or -8. It's easier to understand these ranges if you use a number line and mark the ranges on it. Check the solutions by picking values within each range and checking that the inequality is satisfied. Then pick values outside the ranges and check that the inequality is not satisfied.