32,000 - 100,000 lux is the typical range of illumination that the Sun provides. You don't have to look at the sun, you look at the world it illuminates.
Lux is a "per unit area" quantity - not a "per solid angle" quantity. The variation in values mostly depends on the position of the sun in the sky - when it is low, there is significant scatter of sunlight (most noticeable around dawn/dusk when the sun turns red) which reduces the intensity of the illumination (see for example this earlier answer )
There are three closely related units of "brightness".
First, there is the candela - "the light of one candle". If you look at the light of a 1 cd source on a sphere that is 1 m radius (area $4\pi m^2$), it gives you $4\pi$ lumens. At the surface of that sphere, the intensity of light (per unit area) is 1 lux. If you make the sphere bigger, you will have the same number of lumens (lumina?), but the illuminance (lux, lumen/area) will be smaller.
For reference, a 100 W light bulb has an output of about 1600 lumen; if you wanted a "light as bright as the sun" you would need about 2 kW - and you would have the same illumination at 1 m distance.
Since the total power of sunlight per unit area is about 1 kW (round numbers), that is actually remarkably consistent (especially given the fact that a light bulb is cooler than sunlight and therefore emits less of its radiation in the visible part of the spectrum)