They need the correct date to work properly or they may cause system failures. In the early days of computers, it was necessary to limit the amount of input data as much as possible, in order to maximise processing storage. As a consequence, when inputting years, the prefix '19' was often left out. The Y2K problem comes with the Year 2000 when computers may instead think it is 1900. The doomsayers predicting mass computer failures in 2000 have served the purpose of alerting the world to possible effects that require remedial action now.
Organisations have to allow funds to address and fix the problem.
Fortunately, most systems built since 1996 are compliant but the remedies for others are time consuming and labour intensive. The solutions are proving to be relatively straightforward. An easy way to determine any adverse outcomes is to run operations with test data to see the possible effects. Anyone can do that on a personal computer by changing the date. More subtle changes may be required, e.g. in lifts, or in large systems that may require extensive debugging.