Since Google Chrome version 42, the NPAPI system has been disabled. This prevents older plugins from working with Chrome and is purported to be in order to improve security.
One of the plugins affected by this change is the "Microsoft Silverlight Plugin" which is required for some of the features in Outlook Web App.
Use another browser
The simplest fix for this problem is to use a supported browser. Both Firefox and Internet Explorer support the Microsoft Silverlight Plugin.
Use the OWS Light mode
When logging into webmail, tick the box "Use the Light version of Outlook Web App" which will use a version that does not depend on Microsoft Silverlight.
Enable the NPAPI in Google Chrome (Advanced, not-recommended)
It is still possible to re-enable the NPAPI in Google Chrome as of version 42. This is not recommended, however as Google may remove this feature in the future. In order to re-enable the NPAPI you must point Chrome to chrome://flags/#enable-npapi and click "Enable" under the "Enable NPAPI" heading.
Remember to re-launch Chrome for the setting to take effect.
When accessing OWA, you may need to explicitly allow the plugin to run. Check the right side of the URL bar for a plugin notification.