How are we able to clearly see dust particles in a sunbeam?

We see an object when light reflected from it enters our eyes. There is a threshold level of intensity required for the eye to detect the object. We do not see an object in the dark or in dim light and also a candle kept far away, say at 10 km distance. Extremely small objects like bacteria will not be visible in normal light since the area reflecting the light and hence the intensity is very small. 

Dust particles are much bigger than bacteria but are still quite small for good visibility in normal light. They may be 10-100 micrometres in size. When exposed to light they scatter it in all directions like every other object. A spherical particle scatters light symmetrically around the direction of light propagation (but not spherically symmetrical). It generally scatters more light in the (forward) direction of light propagation. Angular distribution of light intensity depends on the size and shape of the particle, its material properties (for example, refractive index) and wave length of light (0.38-0.78 micrometre for visible light). But the important thing here is that light is scattered in the transverse direction also. It is this scattered light that makes the particles visible. 

A particle may not be visible in the open sunlight but visible in a sunbeam or a projector light beam in a room. In the open sunlight, bright light being in all directions, it swamps the scattered light and our eye cannot distinguish it. In a room the scattered light we are considering is transverse to the directed bright light and the eye can detect it. The eye is also adjusted to the low intensity light. A particle scattering the light, if in motion, also shifts the frequency (colour) of the light. This shift is due to the well-known Doppler effect but is very small compared to the frequency of the original light itself. Hence to measure the velocity of a dust particle by measuring the frequency shift, a very good quality coherent light source like a laser is required. This principle is used in a laser Doppler velocimeter to measure fluid velocity.