Flickr recently removed their WordPress sharing support, so all you get now is some generic HTML code that’s not ideal for WordPress. (Mind you, neither was Flickr’s old sharing code, which didn’t work well out of the box — I described how to fix it in this old blog post.)
Just go to Flickr, find the sharing code, select “Small 240 x X“, then “HTML”, and paste the result into the text box on the left and click the button. The WordPress-style code will then appear on the right, and you can copy it and paste it into your blog post.
It’s not quite as convenient as Flickr’s old system, but it gets the job done.