I guess by train, but it takes some time, keep that in mind. Check timetables and fares here: www.bahn.de
And book rather early in order to be able to get the lowest fares!Edited: 2:15 am, June 10, 2018
Good news, but book well in advance if you want to benefit from a reasonable price!
Unfortunately, this routing incolves some kind of backtracking, making the effort somehow pointless and less efficient.
You could also go from Brussels to Cologne by train and fly from there to Salzburg on Eurowings. But considering the extra time for changing to Cologne Airport upon arrival in Cologne, you will again lose some precious time, why going all the way by train might be easiest after all.Edited: 9:44 am, June 10, 2018