Well I don't know and was bit amaze too that why 5-6 years old most popular social networking service Instagram don't have facility to allow its users to embed stuff on website. But, thanks to Instagram team that, finally on their official blog they released announcement of addition of share button to embed photos and videos on website.
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How to Embed Photos and Videos from Instagram on Website
To share images and clips on a website simply open desired post in a browser. You will see a new share icon (I mentioned it as yellow in above image) close to the comments button. Do click on it in order to get the code. Grab the code and paste it in web page. For adjustments in picture / clip according to web page do change height and width values within iframe. Paste the final code in webpage html. Hit publish to make it live.
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Well its good news for Instagram users and especially for website owners. Because this two click embed feature give them a freedom from headaches and tricks which they were using to grab the similar code from source code.
Note: Share Icon will appear only on public (not private) photos and videos.