Legally, the book is useless to the Bard. How that manifests lore-wise is up to the DM
A Book of Shadow only contains cantrips:
Your patron gives you a grimoire called a Book of Shadows. When you gain this feature, choose three cantrips from any class's spell list.
If that Warlock also has the Book of Ancient Secrets Invocation, they may also add Rituals to the book:
You can now inscribe magical rituals in your Book of Shadows. Choose two 1st-level spells that have the ritual tag from any class's spell list (the two needn't be from the same list).
Given the fact that a class may only read spells from scrolls, etc that are in their class list, this would mean that a Bard, at best, could only read rituals and cantrips which area already on the Bard spell list.
However, the bard would not be able to make use of any Bard spells in the book because a Bard is limited in the number of cantrips and spells they know (as are several other spellcasting classes).
On the other hand, A DM might treat the grimoire as a spell scroll and allow the Bard to cast the book's contents while it was on her person. But this would then mean the Warlock could not then cast those spells because the pact also says:
While the book is on your person, you can cast those cantrips at will.
And the Invocation: With your Book of Shadows in hand, you can cast the chosen spells as rituals. You can't cast the spells except as rituals, unless you've learned them by some other means.
This homebrew solution may introduce some balance issues that cause unintended consequences so that's something to very much be aware of.
At any rate, without the book, those spells are not castable. Once the Bard handed the grimoire back to the Warlock, the Bard would then lose access to those spells. The net number of cantrips & rituals known in the party would remain constant either way (unless the Warlock knew any of the book's ritual spells innately.
How the DM decides to handle the mechanics from a lore standpoint is their choice. The Bard can identify the type of magic for each spell (this is in keeping with the rules of Detect Magic spell) but that's of no real value. The PHB offers this lore explanation:
Or [your Book of Shadows] could be the tattered diary of a lunatic driven mad by contact with the Great Old One, holding scraps of spells that only your own burgeoning insanity allows you to understand and cast.
But maybe the Bard is unable to read the book's writing at all because the text appears as swirling, formless shadows, etc, so the nature of the magic is all she can glean.
There are lots of options for how to explain this limitation.