For anyone new to PlayStation Plus:

The 3 (sometimes 4) monthly games for Essential tier have to be added to your library (via PS web store/PS app/console) before the next batch of games arrive on the first Tuesday of every month, in order to claim them.

You can access the games at any time in the future if your subscription is active and if you had added them to your library before the refresh. If your subscription lapses, the games will show in your library with a lock icon and get unlocked when you re-subscribe.

The Extra catalog of games is different from this as some games leave the catalog and if you had them installed or in your library, you will lose access even with an active subscription when a game leaves the catalog.