"zeitgleich mit dem Erreichen der Flugfläche 100 sprach die Eiswarnanlage (ice detection) an"
The ice detection warning sounded simultaneously (zeitgleich) with (the aircraft's) arrival in FL100.
(Translation note - "erreichen" is to reach, attain or arrive at, "das Erreichen" is the noun form. Arrival sounds weird in this sense, but "attaining" or "reaching" sounds like climbing in English)
"Die Besatzung schaltete daraufhin die Triebwerkenteisung und die Zellenenteisung ein."
The crew switched on engine and airframe anti-ice immediately after the warning.
(Translation note - daraufhin means "immediately afterwards" or "at that moment" but has a sense of causality. In this sense it suggests to me that they switched on the anti-ice immediately after the warning, but the German is a touch ambiguous - "sofort nach" might have been clearer if that was what was meant, or "unmittelbar nach".)
"Die Auswertung der Flugschreiberaufzeichnung bestätigt, dass die Eiswarnanlage angesprochen hat und die Triebwerksenteisung sowie die Zellenenteisung von der Besatzung eingeschaltet worden war."
Straightforward except for the passive construction. "Examination of the flight data recordings confirms (bestätigen - to confirm) that the ice detection warning sounded and that both the engine anti-ice and the airframe anti-ice were switched on by the crew."
"Sechs Minuten nach dem Erreichen der Flugfläche 100 kam es zu Triebwerkschwingungen."
6 minutes after (the aircraft) reached FL100, engine vibration developed.