The Filter core module allows you to configure formats for text input for your site (read more about Text formats and Filters).
"Convert asset tags to markup" filter should be activated to allow Asset's markup being rendered correctly in WYSIWYG and on front-end.
Navigate to Administration > Configuration > Content authoring > Text formats and for each text format that should support Asset input (for example, Full HTML and Filtered HTML) activate "Convert asset tags to markup" filter.
In the Filter processing order group, you need to ensure the correct processing order. Normally, Asset module should proceed with conversion of the markup after the Limit allowed HTML tags filter. So the simplest configuration looks like this:
Note that in the case, when Asset module input will be allowed for non-privileged users and "Limit allowed HTML tags" filter is processed before the Asset module filter, you need to be careful with allowing users to input Free HTML asset, as it might lead to cross-site scripting (XSS). In this case disallow your users to input Free HTML asset via Permission configuration or put the correctly configured "Limit allowed HTML tags" filter after the "Convert asset tags to markup" filter.
If for the one of the Text formats you would like to filter already entered Assets, you need to disable "Convert asset tags to markup" and to enable "Remove asset tags from output". This option can be used for the RSS export of the content, which do not require advanced view of the entered text.