|A particular example|
I assume that you know how to find out URLs of images that are used on a web page.
The URL of the image you see in the tweet is https://pbs.twimg.com/media/A0BRGqGCcAEBdHW.jpg with the dimensions 600px × 428px (scaled to 435px × 310px).
If you click on the image to obtain a larger version you are presented with https://pbs.twimg.com/media/A0BRGqGCcAEBdHW.jpg:large with the dimensions 1,024px × 730px. This still isn't the original image.
I then made a well-educated guess and tried
https://pbs.twimg.com/media/A0BRGqGCcAEBdHW.jpg:orig and it actually worked; I got the original size image with dimensions 2,048px × 1,460px.
So all you need to do to obtain an original size image is to append »:orig« to the URL of the small version of the image.
Note that this only works for images uploaded to twitter not for images hosted on flickr.
Here is one example: https://twitter.com/VirtualAstro/status/241683461155475456 shows a nice photo of a blue moon; the 3 MP image can be found at https://pbs.twimg.com/media/A1qS3JvCQAE0QPv.jpg:orig