Adenosine is not stored in vesicles and released like a neurotransmitter. It appears as a consequence of energy use. When a neuron fires, ion pumps such as the sodium-potassium ATPase work harder to restore ion gradients, and that work consumes ATP. The breakdown of ATP proceeds through ADP and AMP to adenosine, which can leave the cell through nucleoside transporters. Because ATP consumption scales with firing rate and metabolic load, adenosine production scales with neuronal activity as well.
Production is only half the story. Adenosine is also cleared, largely by reuptake and by enzymatic conversion. During wakefulness, production outpaces clearance, so the extracellular concentration of adenosine rises. The longer the brain stays active without a sleep period, the higher that concentration climbs. This is why adenosine is described as a sleep drive that integrates time awake rather than a signal that switches on abruptly.
A useful way to hold this together: adenosine is a running tally of recent neuronal work, written in the currency of ATP breakdown and read out as an accumulating extracellular signal.