Learn English – What does the word “chemistry” really mean here

I think NOAD's second definition of chemistry defines this pretty well:

chemistry (noun)
1 the branch of science that deals with the identification of the substances of which matter is composed
2 the emotional or psychological interaction between two people, esp. when experienced as a powerful mutual attraction

This word is often used when talking about romantic couples, or potentially romantic relationships:

Linda, how come you didn't go on a second date with Rob?
Well, Heather, there just wasn't any chemistry between us.

However, the word also used outside the bounds of romantic or sexual relationships. Two athletes can be said to have good chemistry if they play remarkably well together. A student and his advisor can be said to have good chemistry if they enjoy doing research with each other, and their conversations become catalysts for new ideas. Two actors can be said to have good chemistry if they play off each other very well. For example, in a biography about hockey superstar Wayne Gretzky, Matt Christopher writes:

From the beginning there was an incredible chemistry between center Wayne Gretzky and his right wing [Jari Kurri].

and a CNN piece looking back at the successful Star Trek series mentions:

Leonard Nimoy and William Shatner shared a unique chemistry, making their on-screen performances and off-screen appearances a treat for fans.

In the context quoted by the O.P., however, it sounds like the word is being used mostly in the sense of a strong romantic attraction coupled with compatable personalities.

Whichever way the word is being used, though, two people with good chemistry tend to make each other better.