Why is Young Sheldon's end credit title card numbered #562?

These are called Vanity Cards.

Producer Chuck Lorre includes them in his productions. The Big Bang Theory Season 11 and Young Sheldon Season 1 started on the same day. But TBBT got its show aired before Young Sheldon.

The episode, The Big Bang Theory S11E01 (The Proposal Proposal) has title card number 561. Therefore Young Sheldon S1E1 (Pilot) started with 562.

You can find all of Chuck Lorre's vanity cards on chucklorre.com.

From Wikipedia page about Vanity Cards,

The unique vanity cards for Chuck Lorre Productions have become a "trademark" for Lorre. Typically, on the end of every episode of his productions beginning with Dharma & Greg (an Apple Macintosh computer was used for Lorre's production card on Grace Under Fire and Cybill) , Lorre includes a different message that usually reads like an editorial, essay, or observation on life. A typical card might include a range of topics as diverse as what the Bee Gees never learned, the cancellation of Dharma & Greg, his support of Barack Obama, the competence of AOL Time Warner management, and the genesis of Two and a Half Men.

All Chuck Lorre shows have such vanity cards at the end. You can read all of them on his website.

There's a longer explanation on Wikipedia:

The unique vanity cards for Chuck Lorre Productions have become a "trademark" for Lorre. Typically, on the end of every episode of his productions beginning with Dharma & Greg (an Apple Macintosh computer was used for Lorre's production card on Grace Under Fire and Cybill) , Lorre includes a different message that usually reads like an editorial, essay, or observation on life. A typical card might include a range of topics as diverse as what the Bee Gees never learned, the cancellation of Dharma & Greg, his support of Barack Obama, the competence of AOL Time Warner management, and the genesis of Two and a Half Men.

The full title reads

CHUCK LORRE PRODUCTIONS, #562

I would take this to mean "This is the 562nd Chuck Lorre production". I don't have time to verify this right now, but I would assume that if you counted every Chuck Lorre-produced TV episode prior to Young Sheldon, they would total 561.

Why they're including that number in the end credits in the first place is probably worthy of a separate question, though I'm afraid I wouldn't know the answer.