Short answer: Go to Profile > Settings > Health Data and tap Connect Apple Health. Allow the data types on the Apple Health screen. Dose Track then reads your weight, sleep, resting heart rate, HRV, body temperature, body fat, blood pressure and muscle mass.
Dose Track reads data from Apple Health; it does not send your dose logs to it. The one exception: if you edit or delete a synced weight entry from the Weight tab, that change is made in Apple Health too.
The data is stored on your phone. It only leaves the phone if you turn on encrypted backup, which is locked with your password. The synced numbers feed the Wellbeing page and your weight.
To change what Dose Track can read, open the Health app, tap Summary, tap your picture or initials, then under Privacy tap Apps and choose Dose Track. If you denied access the first time, turn it on here and tap Reconnect Apple Health in Dose Track.
Delete Local Health Data at the bottom of the Health Data page clears the copy in Dose Track. It does not touch Apple Health.