You need to remove the user from any group that has permissions to the project. You can do this by creating a new group for the project and exclude them from the group or you can remove them from the group that currently has access to the project. Keep in mind that the later likely will impact them in other projects.
I would refer you to the documentation. Here is one overview article. It is all quite doable just need to be diligent about how you setup permissions for each project as Joe eludes to. By default permissions are pretty open and when users get added to Jira they typically go into a common group, e.g. developers or jira-software-users and that group is given permissions on each project. It would be a challenge for me to guide you w/o really having access to your instance and spending time assessing things.
First, by default JIRA has a horrible permission scheme that violates security best practices by allowing everyone that can logon to do just about everything.
JIRA works by GRANTING access. You can't restrict access. By default, it grants access to the group used to logon (see Global permissions to see the "can use" groups and admin groups). This is where users are getting their access.
This may be a big effort, but it will pay off down the road by making it easy to control access.
Most of the 'old timers' use project roles. It meets the best practice for security and gives complete control to the project lead for access to their project. JIRA comes with many project roles, but you can add more if you have a special need.
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