The RIO/WSO generally does what their name suggests - they're responsible for the radars and other weapon systems.

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Up until recent advances in automation, handling more advanced weapon systems on combat aircraft was very labor intensive, and generally a dedicated crewmember is added to designs where added systems are particularly labor-intensive. It's a trend we see all the way back to WW2 on night fighters - a dedicated radar operator was generally in the aircraft to direct the pilot, as handling the radar and combat flying at night were both very labor-intensive tasks. While radars had improved in the years between WW2 and the development of guided air-to-air missiles, radar-guided missiles themselves would be a labor-intensive prospect that generally required an extra crewman to use effectively. Single-engined fighters with air intercept radars did exist, but they tended to have the radars for gunlaying and search/track to get them into a position for firing infrared-guided missiles rather than radar-guided missiles. While single-engined fighters mounting advanced radar-guided missiles weren't unheard of in those years, they were rare, and those that did exist - particularly the MiG-23 - tended to suffer from major problems with pilot workload.

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For interceptors in particular like the F-4, F-14, and MiG-31, they're designed with the interception of bombers at standoff ranges with long-range missiles in mind. As such, they've got very advanced and very powerful radars installed and generally the most advanced radar-guided missiles available at the time. In an era before multifunction displays and with limited computing power, that meant that these advanced systems tended to require lots of operator input and take up a lot of real estate in the cockpit. With the systems being so labor intensive, it would generally be up to the WSO to handle all of the tracking, locking, and firing of radar missiles.

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Though two-seat fighters aren't as necessary anymore thanks to more compact and automated combat systems, two-seat strike aircraft persist to this day for similar reasons. In the early days of guided munitions, guided air-to-ground bombs and missiles tended to be very operator intensive, often requiring an operator with a joystick to fly the weapon until it hit the target. Navigation was also a full-time job in more specialized strike platforms, as mission profiles increasingly were relying on very low-level flying or flying in adverse conditions. Aircraft like the A-6 and F-111 were purely strike platforms (although the F-111's particular layout stemmed from the Navy's abortive attempt to have a fighter with side-by-side seating), with the second crewman being responsible for the terrain-following radar and all-weather navigation systems as well as any specialized weapon systems that may be on board. Even the venerable F-15, which in its traditional fighter variant (F-15A/C) is a single-seater, takes on a second crewman to handle air-to-ground duties in the strike version (F-15E).

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Another interesting case study is the F-105 Thunderchief. It was designed from the start as a single-seat fighter bomber, and it was reasonably well suited to that role. But when it was adapted to the "Wild Weasel" air-defense-suppression role, a second crewman had to be added to handle the input from additional systems like the Radar Warning Receiver and the anti-radiation missiles.