Shade is figurative for protection. The tree you choose should be important and highly placed within your organization or in society in general. This expression is close in nuance to “it never hurts to have friends in high places.”
Saikin no gakusei wa gakusha no kenkyuu teema dewa naku, gakusha no chimeido ya seijiryoku ni yotte kyooju o erabu keikoo ga aru. “Yoraba taiju no kage” to you wake ka.
There is a tendency among current students to choose an academic mentor not because of the professor’s research interests but because of the professor’s fame or political clout. I suppose it’s because they say, “Seek shelter in the shade of a big tree.”