The first question isn't what it costs — it's whether your admin has switched it on. These are account settings, not personal ones. Whoever runs your organisation's Zoom account enables them for everyone or for particular groups, and can lock the choice so an individual can't override it. If the icon isn't in your toolbar, there is no setting of your own that will summon it.
Assuming it is on, the core features come with the plan rather than as an extra charge. Zoom's own comparison page says AI is included at no extra cost with paid Zoom Workplace plans, the cheapest being an annual Pro subscription. Zoom's support articles list Pro, Business and Enterprise as the plans that qualify for meeting summary, and don't list the free Basic plan.
The free plan isn't empty, but it is metered. Zoom's pricing page currently shows a small monthly allowance on free Basic — only a handful of meeting summaries, with similarly small limits on the other features. Zoom's help centre and its pricing page don't currently say the same thing about free access, so check what your own account actually offers before planning around it.
First step: host one internal meeting, start the AI yourself, and read the write-up against your own memory of the call. Don't point it at a customer on day one.