The billing section and associated fields will only show up on the core checkout page if an onsite payment method has been configured, or if the product, membership or bundle is paid. If it is a free membership, for example, billing details will not show up.
The Checkout Core Page is a dynamic page. The fields displayed change based on the payment settings configured, whether the person viewing the page is logged in or not, as well as the embedded information provided from purchase links.
If there is an onsite payment method configured and the product, membership or bundle is paid, then you should double-check that the [ MM_Form_Section type='billingInfo' ] and [/MM_Form_ Section] are surrounding the billing section on your checkout page.