The export button on top of the osm.org site is meant to export raw osm data. Since you want to open the map in Illustrator, it doesn't sound like this is what you want.
You can export a map in various image formats (png, svg, jpeg, pdf) from the share buton button on the right hand side (an arrow coming out of a box). An svg image should work in Illustrator, although I don't have Illustartor myself to check. The svg works in inkscape though, and I've used it to edit osm map images before. I've attached a screenshot (from [http://www.openstreetmap.org/#map=15/3.0226/30.9115][1]) to show you how to download an svg image.
This question has been discussed before as the previous answer said, see for example [How do I export map images of larger areas?][2]. There are also programs that can use raw osm data (downloaded using the 'Export'-button) to produce map images with more control. I'm not very familiar with these myself, but have produced images with ![Merkaartor][3] before and heard that ![Maperative][4] is a good tool for this job.
!["Screenshot from osm.org"][5]
[1]: http://www.openstreetmap.org/#map=15/3.0226/30.9115
[2]: https://help.openstreetmap.org/questions/22/how-do-i-export-map-images-of-larger-areas
[3]: http://www.merkaartor.be/
[4]: http://maperitive.net/
[5]: http://help.openstreetmap.org/upfiles/2016-05-26-132029_1920x1080_scrot_Ww74Np8.png
