The reason why nobody is giving you good answers is because Meisner Effect is quite complicated. You need to understand a good deal of quantum mechanics and solid state theory to understand where it comes from and how it works.
The basic statement that arises from this is that magnetic field within a superconductor cannot change without destroying superconducting state. The later is only possible if you excite the superconductor to a higher energy state, so you have to input a certain amount of energy. You can do so by increasing temperature or increasing external field, for example.
From here, all of the properties follow. If you attempt to change external field, the current inside superconductor changes, canceling the external field. So if you move a magnet towards a superconductor, superconductor itself becomes a magnet that repels the one you moved closer. Hence, levitation. You can also pass critical temperature with some external field already present. Then this field becomes "frozen" in the superconductor, and you can later use it as a powerful magnet. Particle accelerators and magnetic resonance spectrometers often use this.
By the way, induction of current in superconductor has little to do with classical Faraday's effect, exactly because the field never changes. Since dB/dt inside superconductor is zero, so is the electric field created by Faraday's Law. The actual process is entirely Quantum, and has to do with effects of vector potential on electron wave function.
P.S. All of the above is valid for Type I superconductor, and not necessarily for Type II, which is a bit different.