I will not go into math (because many already did ) but will try
to explain physics. Wave function describes the wave of
probability (this of course rough analogy) it means when it
travels in space, information of probability (probability of
finding the particle) travels with it, thus this is phase
velocity (which can have higher speed than speed of light) and
it is not a real "physical matter" wave, but just a flow of
information on probability, but it still a special kind of
waves, in sense it can't have additional cargo of informations
that may enable us to pass any additional useful information,
thus it will not violet relativity in any way, while group
velocity describes wave packages speed (also very rough and not
very true analogy, but very helpful to build intuition) and this
package can be imagined as your particle, thus it has the same
speed of your particle. Actually matter wave describes the
probability of finding a particle at any time at any point of
space. It is not exactly a sine wave, it is a