Will SIM card from iPhone 5s work in an iPhone 13?

While the sim card will fit, you should NOT use it. Your very old sim Card is not Provisioned for 5g Service. If you want to use a physical sim in your new phone, get a new 5G sim card from AT&T. OR, honestly, don't use a physical sim card at all. When I ordered my 13 Pro Max from Apple, I identified AT&T as my carrier and verified my cell phone number. When my new phone arrived, the moment I turned it on, it activated immediately to the eSim built into the phone and my phone number was active instantly on the new phone.

Yes, both phones take Nano-SIM cards, so if you use your old SIM card from your iPhone 5s it should work perfectly fine on iPhone 13. If you're planning on switching carriers, I would recommend buying your phone unlocked from Apple. All iPhone models bought from Apple Retail or Apple Online stores are unlocked except for those bought with AT&T Installment Plans. You can still pay monthly for your phone with Apple iPhone Payments or Apple Card Monthly Installments.

Learn which size SIM card your iPhone or iPad uses - Apple Support

If you order an AT&T phone pre-setup for your AT&T account, the account will be set up on the eSIM. Apple announced they were doing that for all iPhone 13 and 13 Pro orders where the buyer provided account information so the device would arrive preconfigured. Since the phone’s have eSIMs, why clutter the world with hundreds of millions additional plastic and metal SIMs? Apple is very public about their efforts to make their products environmentally friendly (or as much so as anyone can for solid state electronics) so not sending physical SIM cards is part of that effort.

P.S. that is also mentioned in the FAQ on eSIMs on the iPhone 13 and 13 Pro order page when ordering online.

If you order or buy other without having it preconfigured for your service, contact AT&T and they will email you a QR code to scan to program your account to the eSIM and enable it for 5G. 5G standards have new provisions for security of your cellular connection that are not compatible with older SIMs.