As others have said, that is sort of the case.
Remember that electric field lines represent the path that a positive (test) charge will follow if placed in the electric field. The proton has a positive charge and so it will move to the right - you can see that towards the right, the density of field lines increases. This means that the field strength is greater and so the Coulomb force on the proton will also be greater (recall
F=Eq). And as we know
F=ma, a greater force => greater acceleration.
A similar argument works for the electron. Hope that clarifies it fully for you
[Forgive my poor notation omitting vectors!]