Manhattan federal prosecutors are poised to open their case Tuesday in the trial of a former software engineer for the Central Intelligence Agency who is charged with handing over a trove of classified information on the spy agency’s hacking operations to WikiLeaks.

In 2017, WikiLeaks released more than 8,000 pages of secret materials—which the antisecrecy organization called “Vault 7”—detailing the CIA’s cyberespionage arsenal, including the agency’s playbook for hacking smartphones, computer operating systems, messaging...