Why can we see the dust particles in a narrow beam of light (and not in an all lighted area)?

Your inability to see the dust until you narrow the slit has nothing to do with the narrowness of the beam but instead the dynamic range of light that your eye can see at one time.

A bit of searching turns up reports of a contrast ratio for you eye at one time as between 100:1 and 1000:1. This means if you're in a room with a range of brightness greater than about 100 to 1 the brightest things will all be washed out as white and the darkest things will all be essentially black. This is obvious in photos that are "backlit" like this one: These horses aren't black but because the ratio of the bright light to the dark horses exceeds the dynamic range of the camera the sky is washed out white and the horses are in silhouette.

Your eye can adjust over time to a huge range but it can't utilize the whole range all at once.

In the case of dust reflecting light, if you allow a lot of light into the room the relative brightness between the small amount of light the dust is reflecting and the rest of the illuminated room prevent you from seeing the dust.

This is fundamental to signal processing. Why can't you hear a whisper in a noisy room? The noise of the crowd obscures the whisper. The difference between the signal you're trying to pick up and the background noise is called the signal-to-noise ratio. In the case of dust, the light let into the room is scattered and reflected in the room and causes the room to be illuminated. This is the noise that obscures the signal from light reflected off of the dust.

Our eyes become more sensitive to light in dark rooms, especially if we give them a few minutes to adjust. Also the illuminated dust in a beam of light shows up because of the contrast between it and the dark background when the room is dark.

The dust isn't really microscopic.

I would partly disagree with other answers here, and say that it is not related to the dynamic range of our eyes, or our eyes adjusting to the light. This is a factor, true, but the main factor is the background contrast.

If you sit in your dark room, with a white piece of white paper in the sunbeam, you won't see the white fluff in front of the bright white paper. It's not a matter of your eyes adjusting - it's more a matter that the background is bright enough to hide the specks, so they don't stand out.

Conversely, if you open up your whole room to the sunlight, so that the specks are hard to see, then hold a piece of black cardboard up, you'll see the specs of dust floating in front of the cardboard, even though the rest of the room is bright, and your eyes are adjusted to the bright room.