It's called 'cloud computing' because when engineers diagram their corporate networks, the symbol that represents the "internet" is a cloud.
The cloud was used as a symbol because it represents a fuzzy area outside of the control of the company's IT people. The internet is like the "vast beyond" to them.
Cloud computing is when companies rent and share computers, storage and apps that are owned by a tech company (Amazon, Microsoft, Google, Salesforce) in another location, and are accessed over the internet, i.e. accessed over the "cloud" part of the diagram.