What is proxy ARP?

Proxy ARP (RFC 1027) is a way to make a machine physically located on one network appear to be logically part of a different physical network connected to the same router/firewall. Typically it allows us to hide a machine with a public IP address on a private network behind a router, and still have the machine appear to be on the public network "in front of" the router. The router "proxys" ARP requests and all network traffic to and from the hidden machine to make this fiction possible.

Consider a router with two interface cards, one connected to a public network PUBNET and one connected to a private network PRIVNET. We want to hide a server machine on the PRIVNET network but have it accessible from the PUBNET network. The IP address of the server machine lies in the PUBNET network, even though we are placing the machine on the PRIVNET network behind the router.

By enabling proxy ARP on the router, any machine on the PUBNET network that issues an ARP "who has" request for the server's MAC address will get a proxy ARP reply from the router containing the router's MAC address. This tells machines on the PUBNET network that they should be sending packets destined for the server via the router. The router forwards the packets from the machines on the PUBNET network to the server on the PRIVNET network.

Similarly, when the server on the PRIVNET network issues a "who has" request for any machines on the PUBNET network, the router provides its own MAC address via proxy ARP. This tells the server to send packets for machines on the PUBNET network via the router. The router forwards the packets from the server on the PRIVNET network to the machines on the PUBNET network.

The proxy ARP provided by the router allows the server on the PRIVNETnetwork to appear to be on the PUBNET network. It lets the router pass ARP requests and other network packets in both directions between the server machine and the PUBNET network, making the server machine appear to be connected to the PUBNET network even though it is on the PRIVNET network hidden behind the router

What is difference between Proxy ARP and RARP?

The address resolution protocol (ARP) is used to associate the 32 bit IP address with the 48 bit physical address, used by a host or a router to find the physical address of another host on its network by sending a ARP query packet that includes the IP address of the receiver.
The reverse address resolution protocol (RARP) allows a host to discover its Internet address when it knows only its physical address.

Why do we type "no IP direct broadcast" in routers configuration

This command stops the client from responding to a possible DOS (Denial of Service)
attack and also stops client broadcasts from bringing up the line