They are Sorcerers with a flexible list of Spells Known.
If we look at Arcanist, under Spell Casting we see:
An arcanist casts arcane spells drawn from the sorcerer/wizard spell list. An arcanist must prepare her spells ahead of time, but unlike a wizard, her spells are not expended when they’re cast. Instead, she can cast any spell that she has prepared consuming a spell slot of the appropriate level, assuming she hasn’t yet used up her spell slots per day for that level.
An arcanist can only cast a certain number of spells of each spell level per day. Her base daily spell allotment is given on Table: Arcanist under “Spells per Day.” In addition, she receives bonus spells per day if she has a high Intelligence score.
An arcanist may know any number of spells, but the number she can prepare each day is limited. At 1st level, she can prepare four 0-level spells and two 1st-level spells each day. At each new arcanist level, the number of spells she can prepare each day increases, adding new spell levels as indicated on Table: Arcanist Spells Prepared. Unlike the number of spells she can cast per day, the number of spells an arcanist can prepare each day is not affected by her Intelligence score. Feats and other effects that modify the number of spells known by a spellcaster instead affect the number of spells an arcanist can prepare.
From this we can see that in order for an Arcanist to cast spells, they first select a number of spells on their spell list as indicated on Table: Arcanist Spells Prepared to prepare for the day. Unlike wizards, they do not prepare this spell in a specific spot. They can then cast any spell they have prepared using an appropriate spell slot, doing so does not expend the prepared spell (they can reuse that spell).
The simplest way to think about it is as a Sorcerer whose list of Spells Known are swapped out each day. When the Arcanist prepares spells, they are selecting their list of Spells Known for the day. Similar to Sorcerers, when an Arcanist casts a spell they use up a spell slot, but still have that spell known available for casting.