I had the same question!

My understanding — relayed through one degree of separation from Daniel Griffin, the dendrochronologist who wrote the piece — is that the core samples are very long and thin, and care is taken not to injure the tree and to allow it to heal rapidly.

Here's a write-up from Carleton with more info about how the tree core sampling process works: https://serc.carleton.edu/trex/students/labs/lab2_2.html