Roughly speaking, it is a measure of how "dynamic" the "dynamic equilibrium" is when ancelectrode is at its equilibrium potential.
Let's say you are dealing with a copper electrode in a slightly acidic copper sulfate solution, a common situation in copper electroplating. If you apply a high positive potential to this electrode, then of course the copper dissolves. Upon measuring the rate of this dissolution, as the current density you are producing, you find that this current density increases exponentially as you crank up the potential, as long as you don't get any interference from mass transfer limitations or other electrode reactions.
Now switch to a highly negative potential. Now copper deposits, and again (in the absence of mass transfer limits or other reactions) the current density varies exponentially with how much you are driving the electrode potential downwards.
What happens at intermediate potentials? The copper dissolution goes along with its exponential rate law and the deposition follows its rate law, but now these reactions are competing against each other so you macroscopically see only the difference between the plating and fissolution current densities. You might have something like $10 \mu A/cm^2$ for dissolution and $1 \mu A/cm^2$ (these are just numbers I grabbed for this argument, not necessarily true kinetics), in which case you see a difference of $9 \mu A/cm^2$ dissolution.
At the equilibrium potential, the two current density numbers generated by the opposing rate laws are the same so you would not see any net reaction, even though in fact both are occurring together at the atomic level. That common amount of current density being "exchanged" between the opposing reactions when you macroscopically see equilibrium, is the "exchange current density".