That is the dry-break disconnect - it is not uncommon for these to fracture.
They screw into the fuel pump assembly.
You don't necessarily have to remove the pump from the tank, but advisable to remove tank from bike in order to have good access to remove the old fitting(s). You will need to drain the tank to perform the replacement installation.
To remove the tank, you need to remove the seats, tail section, front tank cover, then unplug the two dry-break fuel lines and the elctrical pump connection - there is also another electrical connection on the underside of the tank, for the fuel level sender, so be sure to unplug that one also. Lastly on the opposite (left) side are the two vent & drain hoses which you can pull off the nipples. Two rubber mounts at the front & the one at the back & tank will be ready to come out.
The fittings are not easikly removed - that is because they have a sealant on them that is almost like epoxy; even when whole, they tend to snap off when you attempt to unscrew them. You will mostly need to use an EZ-out in order to get them out. Be patient - this may be quite tedious but important you get all the plastic & old sealer out of the threads before installing the new ones.
You will be best to replace the male and female parts as a set - the o-rings may be leaking anyway on the male fittings at that vintage.
There are all metal replacements available here as alternative to the OEM plastic ones
http://www.triumph1.com/daytona.htm
Be sure to note there are three different sets availabe so match the correct year for your '98.
When re-assembling the tail, ensure that the correct (short) screws are used in the front fitting - those go into captive threads in the tank. If the wrong screws are used they can puncture the fuel cell - not good!!
Good Luck