Most explanations of this phenomenon provided in print are very vague and full of analogies, to help shield the reader from very basic simple high school physics. You'll hear things about being "closer to the stall", which might be enough to help you remember the answer, but doesn't really provide a satisfying demonstration. This is my attempt to lay out the answer.

Lift is a function of airspeed and angle of attack. Increase airspeed and you increase lift. Increase angle of attack and you increase lift, until your angle of attack hits the "critical angle of attack", at which point lift begins to decrease again and the wing eventually stalls.

When you're flying straight and level, the wings are always generating exactly enough lift to hold the weight of the plane. So "L = W" here. You're typically not at the critical angle of attack, so increasing angle of attack can increase lift, making "L > W". When you do this, the forces no longer balance, there's now a net force upwards, and any time there's a net force, the plane will accelerate in that direction. We'll see the plane's VSI jump and our track will become more vertical, eventually the VSI will stabilize at some rate of climb and we'll be in balance again. (The plane is only accelerating upwards when the VSI needle is "in motion", once the VSI settles into a particular position, i.e. we're established in a climb or descent or just level, the forces are in balance again and there's no acceleration. Imagine a "perfect" VSI here, ignoring the VSI lag, etc.)

If you put the wing at the angle of attack that maximizes lift, the amount of lift generated depends on airspeed, it will be greater at 120 kts than at 90 kts. If we were straight and level, and then suddenly put the wing at the angle of attack that generated the most lift, creating a net vertical force (and thus a vertical acceleration), that vertical lift force would be greater at 120kts than at 90kts. So if we jerked back on the yoke at 120kts, we're going to create a greater upwards force, and thus a greater upwards acceleration, than if we do that same exercise at 90kts.

So, clearly, we can induce higher levels of vertical acceleration at higher airspeeds, as we can creater higher maximum lift forces. But what else does acceleration depend on? Newton says "F = ma", force equals mass times acceleration, which can be re-written as "a = F / m". This tells us that for a given force F, the acceleration that results will depend on the mass of the object. Apply the same force F to two objects, and the lighter one will accelerate more than the heavier one.

This is the key to Va's variation with the plane's weight (which correlates directly with its mass). Take two identical planes (same wing, etc.), one loaded lightly, one loaded heavily, and fly them both at the same airspeed, say 100kts. When you suddenly put the wings of those planes at max angle of attack, either by jerking back on the stick, or by hitting some turbulence that changes the direction that the air meets the wing, each wing will generate the exact same maximum lift force F, as the variables in the lift equation are the same for both planes (same wing, same angle of attack, same airspeed). But due to its lower mass, the lighter plane will see a greater acceleration result from this force.

Now think of how our planes specify their load limits, it's not a particular force, it's a particular acceleration. We say our plane's load limit is "3.8 positive Gs", that's a maximum acceleration. For our two planes above, if the lighter one has half the mass of the heavier one, when both planes see the same lift force, that can result in the heavy plane accelerating at 2 Gs while the lighter plane will accelerate at 4Gs.

How do we protect a plane from exceeding it's load limit acceleration? The only way is to ensure that we're flying at an airspeed that's slow enough that the lift produced by the wing, when suddenly put at max angle of attack, is small enough that the resulting vertical acceleration is no greater than our load limit. In other words, we have to ensure that the wings can't generate an "F" great enough that our "a = F/m" is more than, say, 3.8g for our current mass 'm'. How can we limit that max lift force "F"? The max lift force will be a function of airspeed; if you slow the wing down, the max force it is capable of generating is lower, so you can ensure the wing isn't capable of generating a force great enough to create an acceleration that exceeds our load limit by limiting our airspeed.

Our heavy and light planes are both flying with the same wings, and so both can see that max lift force if the wing's suddenly put at max angle of attack, but the lighter one can't sustain the same wing forces as the heavier one, as it's easier to accelerate. So we have to limit the max lift force on our lighter plane even more than our heavier plane, and we do this by limiting our airspeed in that lighter plane even more than our heavier plane, i.e. by setting Va even lower on the lighter plane.