His wife was dying.
Of course, the soldiers hadn't phrased it that way. They all took pains to stay out of the courtyard, in which the dragon towered, and it weakly struggled even as maesters stood at bay, and tried what they could.
For this was a war machine for House Lannister, a beast feared the Seven Kingdoms over, and as the Greyjoys had suffered, it was not sleeping idly by.
Lord Tywin stood over the balcony with his lord's face, for he could not go to the beast. He could only watch, for magicks beyond his ken and maesters dragged from Oldtown could not revive the beast.
It surely was at this height of his realisation that the dragon - keeping House Lannister secure, Casterly Rock's gold ever wanted, the bane of his enemies - had now fulfilled its purpose.
Its purpose, as Lord Tywin saw it, to show him the cost of ultimate power.
He felt no misgivings or change in his path. When the dragon sighed, it was not his wife he had lost. He had truly let her body go, and in spirit, this was her vengeful wraith.
Every tool has a task, Lord Tywin turned from the balcony, for it was not in his character to show weakness.