Why is being a commercial pilot stressful?
The stress comes from everything about the job that isn't actually flying the plane.


Dealing with crew scheduling is the #1 cause of stress.  If your phone rings with that special ringtone you assigned to them, you know nothing good is going to come of it.  These calls often end with reassignments, changed overnights, different crew members and sometimes extensions into days off.
Dealing with a stubborn dispatcher who, despite the mess of storms on your route, refuses to give you an extra 500# contingency fuel.  
Dealing with pressure for on-time departures.  This usually comes from a gate agent because they generally are the ones who get the short end of the stick if we push late.  This can be exacerbated by agents who know about certain "tricks" to fool the system into thinking you pushed, which are often specifically disallowed by flight operations.  
Winter operations.  Figuring out where de-ice happens, what special flow control might be in effect and managing holdover times can be a headache, particularly at unfamiliar fields. 
Spring and summertime.  Never a moments rest during cruise but instead a constant look on the horizon and the radar and requesting deviations from every controller you talk to.  These seasons often have days with widespread thunderstorm activity.  These storms for the most part must be avoided and sometimes require constant coordination with controllers to deviate off-route to go around them.  This creates fatigue more than stress, but the two have a way of feeding back on each other.
Holding.  Do we really have the gas to be doing this?  
Diverting.  Coordinating with dispatch, maintenance (if needed), operations at the field, ATC and passengers.  
Timing out.  Hey folks, sorry we diverted but now we timed out, we're staying the night here.
Crew scheduling, again, to get hotel rooms and find out just how bad this day screwed our schedules for the next few days.
That once yearly visit to the AME where you are reminded just how bad your airport junk food diet is and where exactly "hypertension" starts.
Commuting.  It is fairly common that pilots do not live in their base and live far enough away that the commuting means jumpseating or non-rev flying to get to work and back home.  Depending on where you live you can often encounter oversold flights, competition for the jump seats and little backup options for same-day travel (e.g. not flying in the day before your trip).  Not making it work can have consequences depending on how hard you tried to make it work and how good your union contract and chief pilots are.  




Of note, I never found actually flying the plane stressful including operations in low visibility and high winds.  Those are the things that kept the job interesting and I quite enjoyed them.  Having people in the back was also never stress inducing.  As long as I flew safely and kept myself alive, the passengers would end up that way also.  I was mindful of the passengers in selecting alternate cruise altitudes for ride conditions and requesting shortcuts from ATC to make up time but those are not stressful things to do.