It seems Chrome has rolled out an update which no more allows any external site to install extension on your Chrome browser. This update was rolled out to control malicious extension installation. In fact its been tightened so much that even if the extension is available on the stores but hosted from external source it will fail and hence they need to update the Web store & use the inline method to prompt installation.
Said that, there are many users or developers who would like to Install extensions for testing, reviewing etc purpose or even from trusted site they know.
Developer Mode :
Under Chrome settings > Extensions, look for an option on middle right which says Developer Mode. Enable that and you will see two buttons appear, Load Unpack Extension & Pack Extension. You may use the Load Unpack Extension to install manually.
Drag and Drop :
If you have the extension already with you, just drag and drop it on your browser and it will install it. You will have to review the permission to complete the whole process.
Now in case you need to download extension from a website but as chrome is already blocking it, the best will be copy the download link and download it from some other browser and then drop it over extension page.
PS : Make sure you know what extension you are installing and you trust them enough to have it. We will not be responsible for anything that happens because of following this method.