Roughly speaking, 草鞋｛わらじ｝ is a shoe made of straws.

I'll show you the nuance:  
You say "to stand in someone's shoes" which is an idiom meaning "to see from another's point of view; to feel what another feels".
If you stand in your own shoes, and at the same if you stand in someone's shoes, what happens?
You can see from the point of both your view and someone's, right?

So, in Japan, if you wear two kinds of 草鞋 at the same time, you can be engaged in two trades.

Literally 草履｛ぞうり｝を履｛は｝く means to put on *zohri* shoes, and 草履｛ぞうり｝を脱｛ぬ｝ぐ means to take off *zohri* shoes.
From these literal meanings, 草履｛ぞうり｝を履｛は｝く means to *start a trip*, and 草履｛ぞうり｝を脱｛ぬ｝ぐ means to *finish a trip*.

And, 草履｛ぞうり｝を脱｛ぬ｝ぐ means metaphorically to *retire from one's occupation*.  
From the metaphorical meaning of 草履｛ぞうり｝を脱｛ぬ｝ぐ, the metaphorical meaning of 草履｛ぞうり｝を履｛は｝く could be implied as to find one's occupation and start to engage oneself in it, though we don't use this metaphorical meaning usually. 

I'm not sure, but from these metaphorical meanings, it is somewhat natural that to wear *zohri* shoes has a nuance of trade.