Interpretation of the graph of excitation current for magnetic circuit
Why does the current waveform here have such a sharp peak like that?


Because the magnetic core is saturating. If the core wasn't saturating, the current into the device would be a sinewave (assuming the voltage supplied is a sinewave). Here's another example: -



Picture from here. And below is a gif of how this happens due to the core saturation characteristic: -



GIF source.