It is not very clear, but here are two possibilities.
Adults have many responsibilities and meetings and places to be and money to earn and errands to run. It can seem like you never have any time to do anything just because you want to do it. Children, on the other hand, have fewer responsibilities and can spend a lot of time playing and doing what they like.
(That's the perception, anyway. If you asked a group of children if they thought they were free to do whatever they liked, they might complain about having to go to school and do homework and go to doctors' appointments, not to mention being forced to eat broccoli.)
So when an adult says they are "living their second childhood" they could mean they feel like they love their job and their life, and they are able to do what they like to do. Sometimes professional sports players will make a comment to this effect: They are being paid money to play a game!
If it were today, and Mrs. Riley was actively an astronaut, and said "I am living my second childhood," this is what I would think she meant. But that is not the context of the quote, so I think this interpretation is not correct.
The context is a committee hearing, and Mrs. Riley is one of the committee members (she is a US Representative from South Carolina, according to the page marked "II" which is page 6 of the PDF). She is talking about education, and whether astronauts require much, and whether famous aviators required much. She says:
You spoke of educational background or training of one of the astronauts. Looking at me, you know I am living in my second childhood. Did Mr. Lindbergh have a limited education background... High school, I believe.
I do not have a very good guess as to what she means by this, but it might be that she is continuing to learn; that learning is a lifelong process; and that just because you do not have educational certifications does not mean you should be limited in life. She goes on to say
I tried to encourage some of my erstwhile pupils by saying that Mr. Lindbergh didn't finish college and look where he got.