How does a wizard read another wizard's spellbook?

Question

To prepare spells from another wizard's spellbook you have to be able to read it. Since there is no read magic spell (like in previous editions), how does a wizard read another wizard's spellbook?

Answer 1

Simply Reading it.

You can "read" it if it is on your spell list just fine as you can read scrolls of spells on your spell list even if you don't know the spell. If it isn't on your spell list, you should be able to "read" it with something 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 spell book 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.

Casting From It

However, you can't cast it without fully understanding it. Which requires studying it and copying it into your own notation into your spellbook which requires time and gold.

Copying a Spell into the Book. 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.

Copying that 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.

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.

And then preparing it:

Preparing and Casting Spells

The Wizard table shows how many spell slots you have to cast your wizard spells of 1st level and higher. To cast one of these spells, you must expend a slot of the spell’s level or higher. You regain all expended spell slots when you finish a long rest.

Magic Item Spell Books

Some magic item spellbooks might be the exception to that. The DMG lists a few.

• Book of Vile Darkness
• Tome of Stilled Tounge

These are just fancied up spell casting foci that grant spells, benefits or other effects (Book of Vile Darkness can cause insanity for instance).

Answer 2

One wizard cannot directly prepare spells from another wizard's spellbook.

The cost - of both gold and time - of copying spells into a wizard's spellbook represents the time taken to decode the other wizard's personal notation for their casting. For each level of the spell, the process takes 2 hours and costs 50 gp. Once complete, the wizard has translated the spell into their own notation and recorded it in their spellbook.

(PHB, p.114, bottom left box, "Your Spellbook")