Arginine has several nitrogen groups on the same side chain, and these nitrogen groups have lone pairs of electrons. The lone pairs of electrons can stabilize a positive charge when a proton is bound to the proton-accepting nitrogen group.
Lysine doesn't have these extra lone pairs on its side chain, so a positive charge on the proton-accepting nitrogen group in Lysine is less stabilized.
Arginine is better able to accommodate a proton due to the stabilization effects described above, and therefore it is the stronger base. Does that make sense?