What does 'Increased Sign Intensity' mean?

In my experience, Sign Intensity makes the signs stronger, but in separate ways than upgrading the spells via ability increase.

Unlocking Aard rank II and II means that you can cast farther, and affect multiple targets at once.

Increasing the Sign Intensity of Aard gives it a chance to inflict knockdown (the status effect, as opposed to simply knocking them down) in addition to whatever stun chance Aard innately has.

Increasing the Sign Intensity of Igni gives a much appreciated damage boost. With +2 Sign Intensity, my Igni's are dealing about 30 damage per, instead of the lackluster 6 they do with +0 intensity.

I'm not sure what effect Intensity has on other signs, yet.

The effects of sign intensity seem to vary by sign. Quen was the easiest to examine and sign intensity had an obvious effect

Quen

Sign intensity increases the duration of the sign (and therefore also its durability).

For the level II Quen the formula is

duration = 50 + 15 * intensity

for level III Quen the formula is

duration = 90 + 27 * intensity

Igni

The sign intensity significantly increases the damage as Raven already mentioned. My experiments indicate roughly 9 additional damage per +1 sign intensity. The +sign damage and the damage from sign intensity are additive, not multiplicative.