Why does a higher load factor increase stall speed?
What does 'having' a load factor1 mean? It means, by definition, that you are experiencing a force G times your weight. It doesn't matter how it happens: in a turn or loop or turbulence. But in all cases practically all that force (which usually counteracts the weight) comes from lift.

In normal unaccelerated flight, lift = weight. In a more general case, lift = weight $\times$ G. If G > 1, you need more lift than weight.2

How can you get more lift (from the same wing)? Either you go faster, or you increase angle of attack. So inevitably, for a given speed, you'll have to fly at a higher angle of attack, closer to stall. Or in other words, you'll reach stall at a higher speed than normal.

Note that exactly the same thing happens if you 'just' increase your weight, by other means than transient loading with G - say, by having more cargo or fuel. Again, you'll need more lift - with exactly the same consequences.



1 Load factor is applicable in all three axes, but we are implicitly talking about the normal factor in relation to stall speed. When you are accelerating forward or skidding, you'll also have a load factor, but it'll usually be well below 1.0.

2 On the other hand, if G = 0, you don't need wings at all and you'll go ballistic.