It is negative feedback that provides the almost zero voltage across the input terminals of an op amp, but how to understand this with noninverting voltage amplifier maintaining zero voltage across the input terminals (what feedback does)?
A non-inverting amp still has negative feedback - this negative feedback tries to maintain the system in balance and the balance is determined by the op-amp's open loop gain.
For simplicity's sake and for reasonable accuracy, the open-loop gain can be assumed to be infinite. If it is infinite then negative feedback attempts to keep the voltage at the inverting input exactly the same as the voltage at the non-inverting input. If they weren't the same voltage then, due to the infinite gain, the op-amp's output would also be infinity (or hard against the end-stops of one of the power rails).
So, there is a potential divider from Vout to Vin- and from there to ground. If the potential divider were 0.5 (i.e. Rfeedback = Rground) then the voltage at Vout is twice the voltage at Vin-.
This can only mean one thing; that the voltage at Vout is twice the voltage at Vin+.
You can apply a few more scenarios such as if the potential divider were 0.25 (Rf = 3 * Rg) then the output is 4 * Vin+.
As you can probably see, the gain of a non-inverting op-amp is 1 + Rf/Rg where Rf is the feedback resistor from output to Vin- and Rg is the resistor grounding Vin-.