How do EM waves propagate?
Like other waves propagate. IMHO the best way to appreciate this is to shake a rubber mat. When you do this you stretch a portion of the mat, and then the elasticity of the material contracts it back to its original size, but in doing so the rubber is stretched further along. What you then have is a shear wave with speed v = √(G/ρ), where G is the shear modulus of elasticity, and ρ is the density. In electrodynamics the expression is similar, and is written down as c = √(1/ε0μ0) where ε0 is electric permittivity and μ0 is magnetic permeability.
These E-fields produce H-fields and the process goes on.
You can read that light doesn't require a medium, and propagates because the E-field variations induces the B-field variation which induces the E-field variation. This is wrong I'm afraid. Read this:
Robert B. Laughlin, Nobel Laureate in Physics, endowed chair in physics, Stanford University, had this to say about ether in contemporary theoretical physics: "It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed [..] The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum..."
Are the EM fields really moving. My textbooks says it's changes in field that is moving.
Your textbook is right. When you shake your rubber mat the wave moves away from you at say 2m/s, but the rubber mat doesn't. You've still got hold of it.
Do they disappear or continue with their movement.
They continue. The rubber mat doesn't go totally flat the instant you stop shaking.
But I have seen people on this site saying it happens because of continuous induction of Electric and Magnetic Field. How can I relate both of these?
You can't. See Wikipedia:
"Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time".
You need to read up on four-potential. But for now think back to that rubber mat and imagine we're looking at a portion of it. It isn't flat, it's curved, and the slope of this curve at some point is the spatial derivative E, whilst the rate of change of slope is the time derivative B. There's only one wave there, not two. An electromagnetic wave. Check out Jefimenko's equations:
"There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave (electromagnetism). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."
Duh, I have just noticed that this question is two years old!