洗尘: why is a welcome banquet "washing dust"? 洗尘 refers to the traditional Chinese custom of holding a banquet to welcome visitors from afar. However, the literal translation is "to wash dust", which has nothing to do with eating, and I'm also not sure about the connection to travelling.

Where did this phrase come from? Is it referring to accumulation of dust on travellers' clothes, and washing dust is simply part of the welcoming ritual? Or is it something else entirely?
"洗尘"usually used with"接风"，like "接风洗尘"。

it's come from 
明（Ming Dynasty）·凌蒙初(a scholar and offical)《二刻拍案惊奇》第26卷：“虽也送他两把俸金，几件人事，恰好侄儿也替他接风洗尘，只好直退。”（classical Chinese）。

When someone go far,we say "一路顺风"（I wish you a happy voyage），so when visitors come from afar we sad"接风",and after long travel with wind(顺风),there must be dust on visitors。
so we say "接风洗尘"。

送人走时通常说一路顺风，接人时正好就接着这一股风把人接到，所以叫接风！刮风必然会粘上满身的尘土，所以，接风的同时把客人身上的尘土洗掉。