They concluded that there was a dose-dependent relationship between all causes of dying and shorter telomeres. The “p-value” is the estimate that this relationship was due to random chance. That p-value was 0.000000000000002 or 2 in 1,000,000,000,000,000 odds.
In this excellent review of the telomeres of 65,000 Danish people over seven years, they found that shorter telomeres were independently associated with about a 50% higher chance of dying after controlling for all confounding variables.
When data from the Copehnhagen City Heart Study was published in 2013, they concluded that shorter telomeres in leukocytes was not associated with increased cancer after adjusting for age but that it WAS associated with reduced survival from cancer http://jnci.oxfordjournals.org/content/105/7/459.long
In 2015, the same data looked at certain gene tendencies that promote telomere shortening and concluded that cancer survival was IMPROVED when shortening was enhanced in the telomeres.
Contradiction? Not so fast.