The following animation is helpful in understanding the physics of a transverse wave. See notes below.
Although it may not be immediately clear, the above animation will relate to transverse wave motion.
The above animation shows a particle, or object, in vertical simple harmonic motion. It is moving up and down.
For our discussion in this section we will consider a wave train moving horizontally from left to right. Therefore, this above motion, being up and down, is perpendicular to the motion of the wave we will be considering. That is, vertical up and down motion is perpendicular to horizontal left to right motion.
Consider this yellow block above to be, say, a tiny section of rope.
Imagine that you are vertically shaking one end of that rope in order to send a train of waves down it. Imagine the wave train traveling from left to right.
The rope as a whole, though, is not moving from the left to the right. The rope is moving up and down as the waves go down it.
Every small section of rope, symbolized by the yellow box above, moves up and down as the wave moves left to right.
Therefore, the motion of the medium, the rope, is perpendicular to the motion of the wave. The rope moves up and down while the wave moves left to right.
This is the central concept to the definition of a transverse wave.
A transverse wave is a wave with the motion of the medium being perpendicular to the motion of the wave. More carefully, we would say that the direction of the velocity of the medium is perpendicular to the direction of the velocity of the wave.