Whether you're using a basic Fitbit tracker such as a Fitbit One or Fitbit Zip or have invested in a more high-end smartwatch like the Fitbit Ionic and Fitbit Alta, the time on your device is managed the same way—by syncing it with your smartphone, tablet, or computer.
Whatever time and date it is on the device with which you sync your Fitbit, that time copies over to your tracker.
Daylight savings time is notorious for causing havoc with Fitbit trackers as it forces them to record fitness activity for the same hour twice when the period begins and ends. There really is no way around this annoyance, and most Fitbit users accept this quirk as part of the Fitbit experience even though it can mess up their stats slightly.