Why can customers browse my store but not place an order?

Your store is probably still in Demo mode. While it is, the store owner sees an amber Go live button in the admin's top bar (staff accounts don't see it).

In Demo mode the storefront can be browsed end to end, but orders cannot be placed. It exists so you can set up without taking real orders. Click Go live when you are ready to sell; the checklist that opens lets you go live once Payment, Tax and Shipping are set up.

If you are the owner and see no Go live button, your store is already live. Then check that at least one payment method is enabled under Settings → Store Settings → Payment Gateway (an online gateway or Cash on Delivery), and that shipping is set up for the customer's address.