Why can't anonymous users make contributions?
Do you mean an account is required to even fill in your page, or that filling in the page creates an account? 

If the former then it sounds like a permission issue such as Anon not having access to the following


CiviCRM: profile listings and forms 
CiviCRM: access all custom data. 


These Permissions are usually part of default permissions afaik

Within the Profile settings you want to make sure that you have not set Account creation required which can be found in the advanced settings: Administer > Customize Data &amp; Screens > Profile.