"Degrees of separation" means how closely a person is connected to someone else. Here, it means how closely those economists, etc., were connected to James Heckman.
- Heckman himself has zero degrees of separation from Heckman.
- Anyone who has co-authored a research paper or a book, or shared a research grant, with Heckman has one degree of separation from him. (That is one possible definition of connection in an academic/research context. I do not know if that is precisely the definition that your source used.)
- Anyone who has co-authored a research paper or a book, or shared a research grant, with a person having one degree of separation from Heckman has two degrees of separation from him.
- And so on for higher degrees of separation.
This concept became well known in a parlor game in which people try to find the closest path from a given movie actor to Kevin Bacon, based on two people appearing in the same movie. An actor who has been in a movie with Kevin Bacon has one degree of separation from him. An actor who has been in a movie with another actor like that has two degrees of separation from Kevin Bacon, and so on.
For example: Edward Asner was in JFK
(1991) with Bacon. Asner has one degree of separation from Kevin Bacon. Elvis Presley never appeared in the same movie as Bacon, but he was in Change of Habit
(1969) with Asner. Therefore, Presley has two degrees of separation from Bacon.
It has been suggested that everyone on earth has no more than six degrees of separation from anyone else. I don't know if this is true. Facebook has found that, on the average, any random member has about 4.2 degrees of separation from any other member, calculated via Facebook friend relationships.