How is DHCP request message broadcasted?

When your computer sends a packet of data to 255.255.255.255, it will actually map that to the MAC address FF:FF:FF:FF:FF:FF. The packet is then sent out of your computer to the switch (or whatever your computer is plugged in to) with a destination of FF:FF:FF:FF:FF:FF.

Switches know that packets destined for that MAC address should be sent out of all ports so the net effect is that the packet reaches every device connected to the same switch (or a switch further down the chain). It will also reach anyone connected to the same network via a bridge (including bridging wireless APs) as bridges forward all packets including broadcasts (The idea of a bridge is that it should be transparent - everything in one side goes out the other). Routers normally will not forward broadcast messages between separate networks.

This area, that covers all devices connected via switches/hubs or bridging is called a "broadcast domain" in networking speak for that exact reason - All devices inside a broadcast domain will receive broadcasts sent by any other device in that same area.

No it wont traverse routers unless you setup a dhcp relay agent. Broadcast messages are only broadcasted on the broadcast domain of the host (subnet).