In the wizard's class description, it says that
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.
Is the ability to decipher a wizard's spellbook therefore unique to wizards, or would someone without any wizard levels still be able to decipher a spellbook to read what the spells are?
If it isn't in some form of code, you can "read" any spell if it is on your spell list just fine as evidenced by the fact that you can read scrolls of spells on your spell list even if you don't know the spell. Which means any cross over your class has with the book, you'll recognize.
If it isn't on your spell list, and you're proficient in Arcana and the book isn't in code, you should be able to try "read" it with something like an Arcana check.
Arcana
Your Intelligence (Arcana) check measures your ability to recall lore about spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.
That doesn't mean you can grasp the spell from the spellbook enough to cast it. This will tell you what is written in the book. You'd learn what spells are in the book and their general effect.
Think of it like a research paper or college text book of a subject you're not versed in. You can gleam, "Oh, this is a paper on the Solvent Energy of a Protein in a Dialectic Solution", but you don't have the ability to take action on that unless you're versed in that field.
You can't cast it without fully understanding it. Which, for a wizard, requires studying it and copying it into your own notation into your spellbook which requires time and gold. And then preparing it.
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.
For other classes, knowing that it is a fireball spell doesn't allow you to cast it - unless it is a ritual, and you have the Ritual Caster feat or an appropriate class feature, then you may be able to cast it depending on your particular ability.
Some magic item spellbooks might be the exception to that. The DMG lists a few.
These are just fancied-up spellcasting foci that grant spells, benefits or other effects (Book of Vile Darkness can cause insanity, for instance).
Using the Eyes of the Runekeeper invocation, a Warlock can read a Wizard's spell book.
Eyes of the Rune Keeper
You can read all writing. (SRD_V_5.1, p. 49)
Comprehend Languages
For the duration, you understand the literal meaning of any spoken language that you hear. You also understand any written language that you see, but you must be touching the surface on which the words are written. It takes about 1 minute to read one page of text. This spell doesn’t decode secret messages in a text or a glyph, such as an arcane sigil, that isn’t part of a written language.
A potential problem with approach is that a DM may rule that the writing in a wizard's spell book amounts to "secret messages" (though I would not), or be "glyphs," and the text in the PHB about the spellbook refers to a unique notational style for each wizard in their own spellbook.
Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation. (Basic Rules, Your Spellbook, p. 33)
That makes this one less definite as a solution to the problem. Work with your DM on this one. As DM I'd rule in the player's favor for using the spell to do this.
If you come across a spell in written form, such as a magical spell scroll or a wizard’s spellbook, you might be able to add it to your ritual book. The spell must be on the spell list for the class you chose, the spell’s level can be no higher than half your level (rounded up), and it must have the ritual tag.
If the PC is not a Wizard (examples: Eldritch Knight, Arcane Trickster) but chose the wizard spell list for the ritual caster feat for {any number of reasons}, then because the PC can use Wizard spells this feat's feature ought to allow a non wizard to read a wizard's spell book.
The general rule would be that if the player cannot cast the spell in the first place then they cannot make sense of it when they find it in a spell-book formerly belonging to someone else. A DM might make a different ruling if they had a good reason to do so, that's where good roleplaying comes into it.
