In short, by practice. Just going through each scene of the play a few times makes one remember their lines (and all the agreed non-verbal things which there are sometimes way more than there are lines; but, the physical movements also help remember the lines between the movements). Source, having been a minor character in a play.
Also, it is not entirely uncommon to get to adjust the hard-to-remember-for-you lines to something that is more intuitive for you. After all, you are supposed to be in character, so your lines should be something that come naturally for you. And, nobody but your director (and your co-actors) will notice, unless you are Harrison Ford and say “I know” (in Star Wars Episode V).
More prolific actors and actresses also probably have developed their own set of mnemonic techniques to remember the hard to remember lines faster. With good memorisation techniques, and crucially a bit of practice, one can even remember full stories (the base techniques date back to antiquity and beyond, with Cicero documenting his method of mentally walking through a house during his speaches).
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