That’s because the method described by @tomtjes only changes the <img alt="…"> attribute – the text used by screenreaders and other devices that cannot display the image.
What you are looking for is something to change the “Discourse Welcome.jpg” text in your example picture. That is not possible with the Markdown image syntax in the current version of Discourse. You’d have to rename the image on your computer and upload it again, as Jeff said.
But you can use HTML
When you use HTML syntax to embed the image, changing the description is possible without re-uploading, by specifying the <img title="…"> attribute. For example:
<img src="/uploads/default/original/2X/6/621….jpg" width="200" height="60" alt="Test Alt Text" title="Test Title"/>
I cover this technique here in a Discourse manual I wrote, in sub-section “Using HTML”.
The HTML syntax for images was the default in older versions of Discourse and is still possible to use, but be aware that there could be unintended side effects as it is no longer the default. For example, I don’t know if Discourse detects image usage for images referenced in <img …> HTML tags, so that the “clean up uploads” config setting does not accidentally delete such images.