If an airplane is too heavy when it tries to take off, it might not be able to take off safely (it might still lift off, but be unable to maintain flight in the event of an engine failure).
Additionally, the weights of the things that can render an airplane overweight (mainly passengers and their baggage) are not necessarily centered within the airplane's safe center-of-gravity (CG) range; thus, overloading a plane with passengers and baggage could potentially move the CG beyond its forward or aft limits, resulting in loss of control upon liftoff or shortly afterwards.
The absolute upper limit on how heavy an airplane can be and still take off safely is known, conveniently enough, as its maximum takeoff weight (MTOW); this is the maximum weight at which an airplane is capable of lifting off and remaining airborne, even if it loses one engine at the worst possible moment during the takeoff. Note that the calculation of a particular airplane's MTOW assumes that it will be taking off from a long, dry, sea-level runway with a high local atmospheric pressure, engines at full power, a stiff headwind, and flaps and slats fully deployed; when taking off from a shorter runway, or from one covered in slush, or from an airport at high altitude, or with a low-pressure system over the airport, or with a tailwind, or with noise restrictions that require the use of less-than-maximum power when taking off, what matters is instead the maximum permissible takeoff weight, which is less than the MTOW (how much less depends on all of the above factors, along with any others that would make it harder to take off).
As a result, airlines try very hard to keep their planes from taking off overweight. There are eight major contributors to an airplane's weight:
- The dry weight of the airplane itself and of those of its resources that aren't consumed during normal flight (such as hydraulic fluid).
- The weight of the fuel and oil carried by the airplane.
- The weight of the airplane's crew.
- The weight of the airplane's passengers.
- The weight of the passengers' checked baggage.
- The weight of the passengers' carryon baggage.
- The weight of any freight being carried in the cargo hold.
- The weight of other miscellaneous consumables (such as the peanuts and soda, henceforth known as "passenger chow").
Here's a diagram showing most of these contributors:
(Image by mins from this post.)
Of these, the weight of the airplane itself is fixed and unchangeable from flight to flight (unless maintenance and/or repair work has been done on the plane in the interval between flights, which could slightly alter this weight). And the amount of fuel and oil carried is determined by the amount of same needed in order to get the airplane from its source to its destination, fly around in a holding pattern for an hour or so, and then fly to the diversion airport listed in the flight plan, plus a generous margin of safety to account for such things as headwinds, fuel leaks, engine failures, or all of the above simultaneously. This leaves six weights that can be minimised:
- There are only a few crewmembers onboard the airplane, and there is far less variation in physical parameters among crew than there is among passengers (the geometry of the flightdeck and the controls therein imposes hard upper and lower limits on height, and height is fairly strongly correlated with weight); therefore, although the weight of the crew is known exactly and can be minimised, there is little weight that can be saved by doing so.
- The weight of the passengers can be estimated, but, as Therac said, can't generally be measured exactly, for fear of lawsuits by the large of frame. The total passenger weight can be minimised, if need be, by removing passengers from the airplane, but the airlines avoid doing this if at all possible, because taking passengers off a flight costs them money - and they can't try to offset this by charging heavy passengers more than light ones, again for fear of lawsuits.
- Checked baggage can be weighed exactly, and restricting its weight doesn't lose the airline any money (they generally charge passengers based on their number of checked bags, not - up to a certain point, at least - the weight of their checked baggage).
- Carryon baggage - see 5. However, as the weight of the average passenger's checked baggage is much more than the weight of their carryon baggage (16.7 kg/passenger versus 6.1 kg/passenger, respectively, according to this study), there is more weight to be saved by minimising the weight of the passengers' checked baggage than by minimising the weight of their carryon baggage (and, as Therac pointed out, the airlines do care about the weight of your carryon baggage - this is the main reason why they limit the number and size of your carryons).
- Freight (if any) can be removed to lighten the airplane, if necessary (and its weight can be measured exactly), but, as with removing passengers, the airlines don't like to do this except as a last resort, as it costs them money (since now they don't get paid for transporting the freight that had to be offloaded).
- The amount of passenger chow carried is either constant from flight to flight, or else depends on the number of passengers onboard; either way, it can't be independently decreased (at least, not without pissing off the passengers), and the possible weight savings would be minuscule anyway, due to their very small weight.
As you can see, most of the contributors to an airplane's weight either can't be lightened at all (airplane, petroleum byproducts, passenger chow), would incur significant cost penalties in lightening (passengers, freight), and/or wouldn't free up enough weight from being lightened (crew, passenger chow); this leaves baggage (both checked and carryon), and, since the average passenger has more checked baggage than carryon baggage, checked baggage represents a greater opportunity for weight reduction than carryon baggage does.
 As happened to, as previously stated in this comment by Michael Hampton, Air Midwest Flight 5481.