Money was weighed before it was counted. In the ancient world, payments were made by placing metal on one side of a balance until it matched a standard weight. This is why expense comes from Latin expendere — ex- ('out') + pendere ('to weigh'). To spend was to weigh out.
The Latin pendere meant both 'to weigh' and 'to hang' — a balance hangs, and what hangs can be weighed. This double meaning scattered through English in unexpected ways. A pendant hangs from a chain. To ponder is to weigh thoughts mentally. A pension was money weighed out to a retired soldier
The British pound preserves the connection most directly. A pound sterling was originally a pound weight of silver — 240 silver pennies totalling one Troy pound. The currency unit is the weight unit.
The Italian cognate spesa means both 'expense' and 'shopping' — fare la spesa is 'to do the shopping', literally 'to make the expenditure'. French dépense similarly means 'spending'.
The word spend itself is a shortened form of expendere, worn down through centuries of everyday use. Where expense stayed formal and Latinate, spend shed its prefix and became one of the plainest words in the language.