They used to run their own data centres, and still do for the older products. However, when Basecamp 3 came around, they wanted to run it on a more scalable system. When you have your own data centre, you have to administer your hardware. With the cloud, all of that is abstracted and is taken care of your.

Basecamp starter their cloud journey with AWS in 2016, tired Google Cloud for a while, but eventually went back to Amazon.