Why would an aircraft take off earlier than the planned departure time?

Short answer, because they can. The airline makes their own schedule and can change it whenever they want. Departing and arriving early has more upside than downside.

Keep in mind, you must be at the gate at the original Boarding Time. Meaning, yes, if Boarding Time is 10:15, they can start boarding at 10:00 if they want and make last call at 10:16. If a passenger is not present at 10:16, they can close the flight and depart. This is rare, but possible. Point being, the only time that matters is Boarding Time, especially if the flight is delayed.**

There are two notable exceptions to this:

1. Slot Controlled airports where the destination airport gives departure clearance. This is however relatively easy to get moved up.
2. Gate space at the destination.

**In some cases, such as a remote stand, they load the bus once, at boarding time. So if everyone makes it through by 1016 and you're no where in sight. The bus, and your plane, are gone.

Going early when possible can pre-compensate unexpected delays later, which would otherwise delay the plane's next flight. For example, if something breaks down as it's landing and needs unscheduled maintenance.

This could also be related to air crew duty hours.

Duty hours are limited to ensure the crew is getting enough rest to work safely . The hours logged include time spent sitting in the airport waiting for the departure time or on standby waiting to be assigned to a flight. If part of the crew is near their duty time limit, departing early might prevent calling in a standby person to allow completing the flight legally.

If it's near the end of a month in which there have been a lot of weather delays, the airline company may be in a situation that there is no one remaining to call in and their next choice is cancelling flights because they can't legally start them. If they can get the flight off early enough to land within the limited duty hours then they don't have to cancel a later flight.

Weather can be an issue here. If planners are aware of an oncoming storm, it makes sense to get planes out before it hits the airport, or to get planes down a window between weather systems.

In the extreme case of this, you have Delta flight 431, and balls of solid stainless steel from the entire team to put a 737 down the gap between an outer arm of the hurricane and the core.

I run into this quite frequently on the YQR-YYZ (Regina, Saskatchewan-Toronto Pearson, Ontario) route. My strong suspicion is that it's due to frequent delays in Toronto. Sometimes due to congestion or weather, you may not be able to land when you expect and can be delayed in the air while awaiting clearance. Also, the taxiing can sometimes take awhile depending on which runway you use to land, so wind direction and congestion (which affect runway choice) can elongate the wheels-on-the-ground-to-plane-at-the-gate time.

Of course, sometimes wind conditions can slow you down en route, too.

The sooner you're in the air, the more likely you can be at the gate at the scheduled time, irrespective of what's going on at the destination, and when the destination is a busy airport, that's important. (When the departure airport is a quieter one, it seems a lot easier to get into the air early. I've been in the air as much as 20 minutes early flying out of YQR, on jets as large as an Airbus 319.)

So... passengers should expect it. The departure time is an estimated time, but it can change out of circumstance, necessity, and convenience. And of course, as mentioned by others, arriving at the destination early is never a bad thing, especially if you have a meeting to attend or a connecting flight to board. The worst case scenario is that you're stuck in the destination airport longer and the originating one shorter.