I wouldn't use the term "combined speed." What you are asking is the
speed of one as viewed by the other.
There isn't any way that I know of to make it intuitive that the
velocities don't add. The Galilean rule is so grounded in experience. It
comes out from an observed fact, that the light speed seems to be
universal and independent of the motion of the Earth through the
"ether". Thus the Galilean rule for adding velocities can not be right.
Einstein's own writing for popular consumption "RElativity", a modest
paperback if it's still available, is an excellent discussion. IN that
he derives the relativistic law by assuming only that the light speed is
the same for all observers and that two observers in uniform relative
motion are equivalent.
FOr somewhat more advanced students, you could look at Einstein's words
in his autobiography in the Schilpp volume (1949). There he says that he
had realized the following contradiction by the ager of 16: If I chase a
light beam at the speed of light, I should see a spatially stationary
oscillating electromagnetic vibration. Yet such a thing doesn't seem to
exist in nature and deosn't appear to be allowed by Maxwell's equations.
Maxwell's equations are not invariant to the Galilean transformation
because they explicitly contain a velocity, while Newton's laws are
invariant, because they contain only acceleration.