Attackers can quietly move laterally within networks by first gaining initial access, such as through phishing, then using tools and techniques to discover and access other systems on the network. This includes using powershell to run code without touching disks, download payloads from remote systems, and inject shellcode. It also involves using tools like mimikatz to dump credentials and move access from one system to another to gain higher privileges. The goal is often to compromise domain controllers to access domain admin credentials and gain full control.