What is "the core", and why does it need to be clean?
The core is the standard SAP software your company runs. It is clean when nobody has altered it — only extended it from the outside.
SAP ships a large, standard business system. Every customer gets the same one. Almost no company can use it entirely as delivered, because every company does something slightly differently — a discount rule, an approval step, a report the auditors want in a particular shape.
For thirty years the obvious answer was to change the system itself. You could open SAP's own programs and edit them, or hook your own code directly into them. It worked, it was fast, and it is the reason a great many SAP systems today contain tens of thousands of pieces of custom code.
The bill arrives at upgrade time. When SAP ships a new version, it replaces its own programs — and every place where somebody edited them has to be found, understood, and reconciled by hand. A system with heavy modifications can turn a routine update into a project lasting months.
"Clean Core" is the discipline of getting the benefit of customisation without paying that bill: leave the standard software untouched, and put your own logic beside it, connected through interfaces SAP has promised to keep stable.