We use as+adjective/adverb+as to make comparisions when the things we are comparing are equal in some ways.
For an example:
The weather in this summer is as bad as last year
(The weather is equally bad in this summer as it it was last year)
He had to choose a dancing partner. He was as nervous to pick one as
they were of being picked.
You are here comparing between he(man) and they(women). Here, Both the man and women are nervous. Nervous is an adjective in the sentence.
I have broken down your sentences as:
He was nervous to pick a dancing partner(a woman)
The women were nervous of being picked.
Both the man and women are equal in a way as both of them are nevous at the same time. So, the construction you have used in the sentence is very comman in English.
There is always another way of saying the same thing differently, so, you could also write it as follows:
He had to choose a dancing partner. He was as nervous to pick one as they were nervous of being picked.
He had to choose a dancing partner. He was nervous to do so and they were also nervous of being picked.