What do the Airway Letters and the following numbers mean?
RCTP - Taipei, Taiwan Airport
AJENT SID - Standard Instrument Departure leading to the Navigation fix named AJENT 
MKG - fly to the MAGONG VOR Navigation beacon
A1 - follow airway A1
ANPOG - until the fix named ANPOG
R200 - follow airway R200
GLN - until the GUANLAN VOR Naviation beacon
W22 - follow airway W22
IDUMA - until the fix named IDUMA
IDUMA STAR - follow the IDUMA Standard Arrival Route
ZGGG - land at Guangzhou Baiyun International Airport 


A1, R200 and W22 are airways. They are defined by lists of waypoints. The
waypoints between the first and last used on the airway are omitted and replaced by the airway name, otherwise the flightplan would be extremely long. 

As an analogy one could describe a route on the road. For example Frankfurt - Amsterdam could be described as: 

FRA - 44 - FRA KRZ - A3 - BORDER - A12 - KNP OUDENRIJN - A2 - KNP AMSTEL- S110 - AMS

Without mentioning all the highway intersections and exits, the route is described in sufficient detail to drive it. The numbers of the highways appear to be somewhat random, but I am sure somebody has thought about how to best assign the numbers.