You can embed Mastodon posts in your website or blog by doing the following:
1. Log in through your server's website
2. Go to the post or reply you want to embed and click on it to show it in an enlarged view
3. Click ⋯ on the post and select "Get Embed Code"
4. Copy-paste the HTML code into your website or blog
Lots of common questions answered about this at:
https://fedi.tips/how-to-embed-mastodon-posts-on-a-website
If you have problems embedding, see the article for suggestions.