Which information do we get from magnitude and phase spectrum?
You get the exact same information you will receive if you analyse a 1-D signal using the Fourier analysis tools, for example. To illustrate this consider the following examples



We perform Fourier Transform on it and obtain the following spectrum



As you can see there are two symmetric dots representing the frequency that is present in the image and the central, DC, component.
Next up



Again the Fourier Transform



It is obvious that the higher frequencies are further away from the DC, central, component. To understand the logic behind this you shall remember that we are actually working with complex numbers and their properties kick in.
To answer your second question: You are usually working with sparse data structures (vectors, arrays, tensors, etc.) in the frequency domain. Now you tell me which is easier: Working on the original data set consisting of 500 different values or the transformed one with half of them 0s ?
Now, the importance of phase - drum rolls - it is the same, meaning HUGE
Again, a sample image



Fourier Magnitude Spectrum



Fourier Phase Spectrum



Now, we inverse the transformation by using just the amplitude and then just the phase information
Inverted amplitude spectrum



Inverted phase spectrum



I think you get the idea :D
Now, here are some sample images and their respective amplitude and phase spectra, so you can practice a wee bit