What is the purpose of several mandatory stops (RWY AHEAD) at busy airports?
A full stop at a red sign is only mandatory if ATC has not given you explicit clearance to proceed further. Aircraft will be instructed to "hold short" of the runway, and these signs indicate where the aircraft should stop.

The CAT II/III signs are only for holding when CAT II/III operations are active on that runway. This refers to categories of the Instrument Landing System (ILS) that are currently in use. An aircraft only needs to stop at the sign applicable for the current conditions, as instructed by ATC. Generally if an aircraft must stop at the CAT III holding point, they will later be cleared either to take off or line up and wait on the runway. That clearance means they do not need to also stop at the CAT I point.

On page 47 of this ICAO Aerodrome Standards manual there is a table showing the required distances from runway center line for holding positions. For example, a long runway (code 4) would require 75 m normal distance, and 90 m for CAT I/II/III approaches. These numbers increase with runway altitudes over 700 m (2,300 ft) elevation. There are also many other factors in the airport configuration that might require a greater distance.

The different holding points are to ensure that in lower visibility conditions, there is more room for added safety. It also helps to ensure that holding aircraft do not interfere with the precision approach signals. The CAT III holding point is further from the runway to ensure that the guidance equipment will provide CAT III accuracy and avoid interference from aircraft and vehicles. If there is not a clear procedure in effect to keep these areas clear, interference can happen, and the faulty signals can result in the airplane following an incorrect path.