The reason is, you are using a 7+ year old OS X and the Safari browser you are using is outdated. That is why you only see basic email interface. To get the modern version, you have two options in here. You can either install another browser like Mozilla Firefox (which will support on your Mac for a while) or you upgrade your OS X to latest 10.11 El Capitan.
Option 1 :
Download and install latest Firefox from here --> https://www.mozilla.org/en-US/firefox/new/
After downloading Firefox, open the downloaded file and click and drag the Firefox to Applications folder. Then open Firefox from Applications folder. Firefox is a browser like your Safari. Open Firefox and try to access Yahoo on it.
You can upgrade the OS X to 10.11 El Capitan if your Mac can handle it. Upgrading to El Capitan also update Safari and other apps, thus you will get the best version.
Follow the steps below to upgrade the OS X if your Mac support it.
Click on the Apple icon and App Store. This will bring App Store. Click Updates tab. If your Mac can handle latest OS X, then you will see Mac OS X El Capitan there with FREE UPGRADE button. Click Free Upgrade and enter your Apple ID and password.
This will download the update. It may take a few hours to download. Once the download is completed, it will bring the OS X Installer automatically. Follow the onscreen instructions to install it. It may take up to 3 hours to install the upgrade.
If you have any difficulty on following the steps above, please let me know and I will continue to help you