Cookies are just files. Chrome and Edge keep them in a SQLite database under User Data\Default\Network\Cookies (older builds used \Default\Cookies), encrypted with DPAPI on Windows and the OS keychain on macOS; Firefox stores them in cookies.sqlite inside each profile folder. An attacker with local code execution can copy that file directly and run a scripted SQLite query against it, or drive the browser's own decryption: infostealers like RedLine and Lumma bundle a SQLite reader plus the DPAPI unwrap calls needed to turn the encrypted blob into a usable cookie, while a malicious extension holding cookies and host permissions can read live cookie objects through chrome.cookies.getAll without touching disk at all.
A second mechanic skips the database entirely and pulls cookies out of a running browser process. Chromium exposes a remote-debugging interface over the DevTools protocol when launched with --remote-debugging-port; tools built on that interface, including forks of evilginx2 and dedicated cookie-grabber scripts, attach to the port, call Network.getAllCookies, and receive every session cookie already decrypted by the browser itself. This is the exact mechanism Elastic's detection rule and the SpecterOps and embracethered.com research it cites both target.
However the attacker gets the cookie, the payoff is the same: replay it. The stolen value is loaded into a fresh browser profile, a proxy tool, or attached directly to HTTP requests against the target service, and the session continues wherever the legitimate user left off. No username, password, or MFA prompt is needed, because the identity provider already trusts the token that comes with the request.