Traditionally you have had larger computers called servers that your organizations did most of their computer work on and the data was stored on them. These servers were in the computer room at your organizations. The organization had to buy powerful enough machines to handle things at the highest demand in the organization and plenty of storage that was needed when the most stuff was stored. However, for much of the year that excess power and storage space sat unused, so you were paying for stuff not needed then.
The cloud is where your computing and storage is kept in storage and computer areas somewhere kept by companies like Amazon Web Services (AWS), Google and Microsoft and they provide you with the computing power and storage space as needed.
Serverless computing is the term used in this new way of computing. The organization is no longer having to get servers to run their computing needs. Instead as they need computing machines, they simply use the equipment on the cloud (still on servers) for the amount of equipment they need at that point. When demand goes up in their organization for computing power, they pay for more power and storage as needed. Then when the opposite happens, they spin down the extra they were using, and they only pay for want was needed.
Since you are not having to keep track of space and power on specific servers and only getting the amount needed, the term serverless computing was created. You are sharing the servers at the cloud provider with other organizations and keeping the servers running and having the storage space is no longer the organizations requirement but is now the cloud provider responsibility. The cloud provider must pay the cooling costs and the physical space where they keep these machines and rents space and power.