Good evening, Woodelf.
The 12 month eligibility period only applies to contribution-based ESA, which is not means-tested and is paid as long as the claimant has made enough National Insurance contributions.
When a person is not eligible for contribution-based ESA (whether that's because they've not paid enough National Insurance or because their period of entitlement has expired) they can be considered for income-based ESA. That is means-tested. In order to qualify for the full payment (which is the same as for contribution-based ESA) the applicant must have savings of less than £16,000 and their partner (if any) must not be employed for more than 24 hours per week. If those conditions are not met the amount paid will either be reduced or non-existent.
So many people notice no real difference when their entitlement to contribution-based ESA expires. They continue receiving the same payments. But some people (with partners in employment, or with substantial savings) might get nothing at all.
Chris