ISO 9001 requires records to be kept on certain activities. The phrase to look out for in the 2015 version of the quality management standard is documented information shall be retained.
Aside from auditing and certification purposes, keeping records provide important data for making fact-based decisions on changes in your business.
Here's the list of clauses that require you to retain documented information (i.e. keep records):
Clauses with the phrase "to the extent necessary" or "as applicable" allow some discretion for you to decide what records are needed.
There are couple of other clauses that require you to "monitor and review information":
which will be difficult to do without having documented the information in some way.
During an audit, these records provide necessary evidence for your auditor to determine whether you satisfy the requirements of the standard. Due to changes in required documentation for ISO 9001:2015, auditors are now much more interested in what you actually do (records), rather than what you say you do (procedures).