Since I feel this question was not clearly answered, I am rewording it here.
How does energy reaches from left to right in a transverse wave when the vibration of particles is only in "up and down" direction?
I'm visualising it like this: the first particle gets the energy, it gets excited and jumps up. Then it transfers the energy to the second particle and settles back. The second particle now gets excited and jumps up and transfers the energy to the third one and so on. Am I thinking in the right way?