Why westerners like to wear shoes inside their house?aren't they dirty after wearing them from the outside?

It depends upon the house. Whenever you enter somebodies home you usually want to ask whether or not you should take them off (or just watch to see if the other person does).

We aren't dirty, we just have different costumes. Most of the time shoes aren't overly dirty and if they are dirty you are going to want to take them off anyways (nobody wear's muddy or wet shoes inside). Plus most people vacuum their homes regularly.

People in many other countries, including many in Europe plus Britain also don't remove their shos inside. It's not a matter of cleanliness but of convenience for them.

I'm from Denmark which is considered a western country and everybody takes off their shoes when they're inside their house.

obviously the same need to remove shoes doesnt exist in the western world as in other cultures but certainly there is still an appreciation of not wishing to track in mud or wet

perhaps the difference lies in how the home and floor is used in the west

all our furniture sees us sitting or lying no closer to the floor than say 450mm

as to cleanliness generally, most people regularly, sweep, vacuum and mop the floors of their home

Yes, we wear our outdoor shoes indoors. We weren't brought up not to.