The king rejoices in the Lord because of the power which the Lord has shown on his behalf. He rejoices at the salvation he received. You can think of a rescue from a situation of emergency. The two parts of the sentence are parallel. The second part of the sentence explains the first part a bit more. In the second part of the sentence, we read what the effect is of the power and the expression of joy mentioned in the first part.
1 O LORD, in your strength the king rejoices, and in your salvation how greatly he exults!