Forward converter magnetizing current and active clamp
Can somebody explain why the magnetizing current starts at a negative value?

It starts at a negative value because it finished at a negative value the previous cycle, and does not change instantly.
These waveforms are drawn assuming the converter has been operating for a while, the so called steady state. At first switch-on, the current starts from zero, and will swing up to 2*Im during the first half cycle. What happens next to get to the steady state is more complicated than simply being in the steady state, which is why treatments like you're reading ignore it.
The resistance of the primary generates a voltage across it which tends to reduce the effect of this switch-on transient. The higher current may or may not saturate the core. If it does, a very large current will then be drawn, which will settle the converter down to the steady state faster. Iron cores tend to be run close to saturation, so are likely to saturate on switch-on. Ferrite cores tend to run below saturation, so may well not saturate on switch-on.

I was taught that capacitor voltages can't instantaneously change. Also what does having large value of this cap have to do with the voltage being constant?

You were taught correctly. The capacitor voltage does not jump discontinuously, changing only as current flows in or out. The larger the value of the capacitor, the less the voltage changes for any given change of charge. As the capacitor becomes very large, the voltage change at its terminals becomes very small, eventually changing little enough to be called 'constant', at least constant with respect to the other voltage changes happening in the circuit, and over the timescale of one switching cycle.
For some reason, the author has chosen not to graph the capacitor or inductor voltage, perhaps thinking it's obvious. It is for him, but not for all of his readers. What will happen is that the voltage on the capacitor will adjust itself over a large number of cycles so that the average voltage across the primary is zero.