That is short for "Follow the instructions". The whole process of upgrading firmware is referred to as "flashing", because the non-volatile RAM is also called a "flash ROM".
The long instructions would be to click on the "choose" button, then use the dialog box or whatever they call it that appears to select the file name for the firmware binary. The HyperWRT binary I have, for example, is named "HyperWRT_2.0b4_G.bin". Once you find it, if you click on the file name, causing it to be highlighted, you can click on the "open" button to get rid of the dialog box. The box for the file name is shorter than the file's name, so you might not be able to verify that you have the right one. Go to the bottom of the page and click on "upgrade", which starts the "flash" process.
It warns you not to stop it during the process. If it does stop there might be ways to recover, but you *don't* want to learn how to do them! The best policy at that point is to move back from the computer about two feet, and guard it from kids, cats dog, humans, falling trees and yourself. Don't touch the keyboard and threaten anything that might with death.
It takes a little more than a minute to do with a 3+ meg file. And then it takes a few seconds while the WRT54G reboots. You can watch the front panel lights to tell when it is finished booting. At that point it's back to "default" settings, so you'll have to access it at 192.168.1.1, and if you do that from the LAN ports you'll need to use DHCP to configure the host NIC, or use as static IP address in the 192.168.1.100 to 192.168.1.50 range.
I would highly suggest looking at the Sveasoft firmware too. It is significantly more developed, particularly for those who like to use the web interface. HyperWRT is okay if you want a minimal system and don't intend on downloading the software and doing any development on your own.
My biggest complaint Sveasoft's Alchemy v1.0 is that it does not allow mounting nfs filesystems. Next was that is has a vi but no emacs. And third is that the telnetd implementation doesn't allow turning off software flow control. The lack of nfs is serious, but the other two are of little significance to most people. (I've added those to mine, but it was not trivial.)