Voight wanted Woods’ word that Olinsky would be freed if Voight told the truth about what happened to Bingham; Woods agreed, and Voight prepared to spill his guts.
It was enough to break Voight, sending him to Woods to set the record straight.
Unfortunately, his guts weren’t the ones in immediate danger, as Olinsky was back in prison being shanked in the belly.
Voight was understandably wracked with guilt that Olinsky was in jail for a crime that Voight committed, and he wanted to get his friend free more than anything.