I haven't seen the movie but it is based upon the real life case of The West Memphis Three. As your quote shows Damien Echols was sentenced to die but the death penalty in the USA is a long and complicated process.
When someone is sentenced to death in the United States the case is automatically sent for a direct review in state appellate court to ensure that the original court's decision followed all legal procedure. The court can rule that either (1) the sentence stands, (2) the sentencing court made legal errors and a retrial is ordered, or (3) that no reasonable person would have given the defendant the death penalty and will order the defendant acquitted of the death penalty but sentenced to the next most severe punishment for which the crime is eligible. This process can take a while but if the death penalty is affirmed it doesn't end there.
A defendant can file other appeals in their case known as a federal habeus corpus. This moves the case out of the state courts and into federal courts. These appeals usually claim a violation of civil rights or attempt to admit new evidence. For more on this you can read the wikipedia article on Capital Punishment in the US.
The average time a defendant in the USA spends on death row is ten years but some can spend twice that according to Death Penalty Info. For the record, the longest time an inmate in the USA spent on death row was 40 years. Gary Alvord was sentenced in Florida on April 9, 1974 and he died on May 19, 2013. His execution was delayed because he suffered from schizophrenia and psychiatrists refused to treat him just to make him well enough to be executed.