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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.
@FediTips I find this problematic: Today I embed a Toot in my website, tomorrow the content of the Toot is changed and my website shows something totally unrelated, or even contradicting.
In order to embed a Toot, I need a hardlink to a version which does not change anymore. Otherwise it's a screenshot for me, and a link to the Toot.
If you also boost the toot from your Mastodon account when you embed it on your site, you will be notified in Mastodon if the toot changes.
@FediTips This is a workaround, yes. I don't necessarily want to boost every Toot I include in a blog posting though, especially when the content is negative.
There have been enough news postings about embedded Tweets which later got edited and showed something completely different.
Appreciate the problem, just not sure how else it could be done 
If you allow editing and embedding, then there will always need to be some kind of account involved to notify you about updates to the post.
(It's sort of similar to posting links on the web in general, the site you link to may change after you publish the link.)
@FediTips the difference being if I add a link, it's just that, a link. If I embed a Toot, the content becomes part of my website.
This was abused in the past, once Twitter allowed editing Tweets. One must be aware of that.
It's a fair point, it is indeed important to be aware of the risk. I added a section to the guide as a result of your reply.