Videos don't stream when I click on the list item that holds all my videos in SharePoint 2010

I think it depends on IIS MIME settings, how is the particular video filetype served.
So, try to look in the inetmgr application (start, run, "inetmgr" without quotes), click your local webserver name and choose mime types icon on the right pane. Then in the list, find the extension of your videofiles and check the settings. If it is application/octet-stream, then the file is sent and not played. Check some other video filetype settings and compare it.

My video type is .wmv. I have checked the setting for this in IIS and under column 'MIME Type' the value mentioned is 'video/x-ms-wmv'. Same setting is for all the other video files. This still doesn't work.

Seems then it is a browser issue, rather than server. What browser do you use?

IE 8

Just to let you know that clicking on the thumbnails in the default view (/Forms/Thumbnails.aspx) plays the video. However while clicking on thumbnail in the view (/Forms/VideosListView.aspx) that I created, starts downloading the file.

Okay, so that seems like it the code in your view is the cause. Right click on the link in default view and choose properties. Where does the link point to? Compare it to your view.

Yeah you are right. The one that is playing video is pointing to:
http://ukosp2010wfe01/_layouts/images/VideoPreview.png

and the other one is:
http://ukosp2010wfe01/_layouts/images/icwmv.gif

Can you please tell me how this can be changed. I went into the library settings and tried to find out something there but no luck.