To download the vSphere Client for ESXi 5.3 or 5.5, you'll need to visit VMware's official site or search for the exact version in their archives, as those older clients aren't readily available anymore.
The vSphere Client is used to connect to and manage either ESXi servers individually, or more often to connect to a vCenter server to manage multiple ESXi servers in a single interface (or "pane of glass").