An energy source will supply a constant current into the load if its internal resistance is:

For a source with EMF $E$ and internal resistance $r$ connected to load $R$, $I=\frac{E}{R+r}$.For $I$ to be constant (independent of $R$):If $r\gg R$, then $R+r\approx r$.
$$I\approx \frac{E}{r}$$
In this case, the current is roughly constant even if the load $R$ varies slightly. Thus, a constant current source has very high internal resistance.