There is no mention of shoes among Aaron’s garments – the priests serve barefoot.
One reason for the requirement to walk barefoot in the Temple is that it allows a person to make bodily contact with the physical essence of the Temple lying under his feet. Shoes would be an interposition from this perspective, and must therefore be removed.
Wearing shoes also indicates independence. A person wearing shoes can move freely, wherever he wants and on almost any surface, whereas a person who walks barefoot is significantly limited in his movement. Thus, by removing his shoes in the Temple, a person renounces his independence, submitting instead to the Temple service, its solemnity and its strictures.
These two aspects – physical closeness to the Temple, and renouncing one’s independence there – are interrelated, since, in principle, any closeness in some sense limits a person’s independence. For that reason, wherever there is holiness God says, “remove your shoes from your feet” (see, e.g., Exod. 3:5 and Joshua 5:15). By doing so, one expresses his feeling of closeness to God, and his willingness to submit to God’s will.