Metaphorically, shoe leather cost is the cost of time and effort (or opportunity costs of time and effort) that people expend by holding less cash in order to reduce the inflation tax that they pay on cash holdings when there is high inflation.
[1] The term comes from the fact that more walking is required (historically, although the rise of the Internet has reduced it) to go to the bank and get cash and spend it, thus wearing out shoes more quickly.
In a period of high inflation, people are discouraged from holding large amounts of cash because its value deteriorates quickly relative to the rising prices in the economy.
People tend to hold most of their money in a non-transactions bank account and keep only very small amounts of cash with them.
This causes them to make regular trips to their bank to withdraw cash to pay for goods and services.