To get your own buttons, just go to AddThis social bookmark, click on GET YOUR BUTTON and that will take you to a page as shown in screen shot below:

select the type of button you want, chose your blogging platform which for the majority of readers of this blog will be BLOGGER, then click GET YOUR BUTTON. If you do not already have an account, sign up for an account. You will then be taken to a page where you will get the codes for adding to your template which in my case will be as shown in the scroll box below (yours will have a different addthis_pub=""):
To include the button beneath each post, go to LAYOUT, click EDIT HTML, tick EXPAND WIDGET TEMPLATE, search for <div class="post-template"> tag and paste the code immediately after the tag. A fast way of finding the tag is to press ctrl+F and search. After you have pasted the code into the template, PREVIEW and if everything OK, save template.
IMPORTANT: Always backup template and backup Widgets (formerly called Page Elements) before editing the template.