There are multiple reasons for the width observed in a spectrum.
First of all there is always a width simply due to the precision of the measurement. If the measurement was taken with poor resolution, you will get a broad spectrum even if the underlying molecule has a stick spectrum. In that case the width is not a feature of the molecule but of your measurement apparatus and the measurement has only a very limited information content. If the resolution is bad, the sticks merge and you get one broad peak instead. You can determine if that's the case if you do the measurement with higher resolution. The peaks will get more narrow and your broad peak will separate into multiple peaks with smaller width.
Besides that, there is also a natural limit for the width of the peaks due to the limited lifetime of excited states. Mark Mitchison answer addresses that point very accurately. The lifetime of excited states limits the width of peak to a minimum and you can't decrease the width of peaks below this limit no matter how good your setup is. This width is a property of your molecule and allows you to determine/define the lifetime of an excited state.
Besides this, we have broadening due to the environment of a molecule and simply the fact that we typically measure a whole bunch of molecules at once and not a single lonely one. Measurements of single molecules/atoms are possible but require advanced experimental techniques. These measurements look alot more like stick spectra that are known from atoms but typical measurements of organic molecules are done at room temperater in solution. In in that scenario environment/solvent effects have a large effect on the shape of the spectrum.
In the shown example i would assume that the measurement was done at room temperature in solution. In that case, every molecule has a different local environment which affects the energy of the states contributing to the absorption spectrum. The broad spectrum that we see can thus be seen as sum of many(lets say we have a concentration of one milli mol ~ 10^20 molecules) slightly shifted spectra. This summation will results in a very broad spectrum even if you do have good precision for single given spectrum. Just imagine the sum of 10^20 gaussians with small width spread around a central frequency, the sum will be one broad gaussian if the spread is noticeable.
Assigning peaks to states is thus not straight forward especially not in molecules were you have electronic states, vibrational states, rotational states, spin states ... Most of the states get mashed up under one broad signal and can only be separated clearly into single peaks if the experimental setup is carefully designed to do so.