Most of the modern browsers read malayalam unicode fonts properly. If you are using an older version, please try an upgrade.
If you have problems with the proper visibility of the site, or if you don't want to upgrade now, then the next thing to do is to download Meera font and install it on your computer.
Click here to download Meera unicode font. Open the file and just click on the "Install" in Linux and Windows 7 or newer systems. Refresh the browser and you are ready. The Windows XP diehards may have to go to Control Panel and paste the file in the "fonts" directory. See this page for malayalam resources for Mac.
For any issue with the malayalam fonts or for enabling malayalam in your computer, you may refer this page.
Credits - Meera unicode font is designed by K.H.Hussain and Suresh.P. (Suruma Suresh) (Swathanthra Malayalam Computing)