How does the negative feedback actually feeds back the voltage to the inverting terminal
Roughly speaking, you're describing the effect of negative feedback, for which the intuition is that it brings/forces the system to an equilibrium point.

The reason why the two voltages at the input terminals are essentially the same (putting aside several practical factors) is in part the negative feedback, but mainly the fact that the gain of the amplifier is extremely high.

The simplified/ideal model of the op-amp describes it as having infinite gain. So, if the output is not driven to saturation, then a finite output voltage must mean a zero input voltage (i.e., zero difference between the two input terminals).  In reality, the gain is not infinite;  it is just extremely high;  thus, the difference between the input terminals is not really zero; it is just extremely close to zero.

The negative feedback plays the role of maintaining the system inside its non-saturating / linear behaviour, which combined with the above makes the input differential voltage essentially equal to zero.