Firstly you need the kernel sources and a C compiler(GCC), you can install them from your redhat CD, and make sure the kernel sources matches your current running kernel(if you ran up2date to update the kernel you need to update the sources as well).
Then you should be able to compile the driver by running "make" in the directory, a r8169.o will be generated and you can then copy it to /lib/modules/<kernel version>/kernel/drivers/net
run "/sbin/depmod -a"
put "alias eth0 r8169" in /etc/modules.conf (or eth1 if its the second NIC)
and modify the ifup scripts (or use the network configuration tool) to activate the device.
If it didnt compile for some reason, take another look at the readme or the Makefile, maybe there's some variables you have to modify yourself (like location of kernel sources etc)