When editing your story in Twine, click the story title at the bottom left and select "Publish To File". Save the story somewhere on your file system (other than in the story folder of Twine). It will create a new .html file which can be placed on a website where anyone can play it, assuming you link to where it's stored on the web server. If you need a free website, a great place is CloudAccess. I've never done the following, but there are sites where you can share your story with communities, such as TextAdventures.
In the bottom left of the twine editor, there should be the name of your project with a little arrow next to it that points up. If you click there it will give you a menu, in which you can publish your story to an HTML file. From there you can either send it directly to people to open up on their computer or host it somewhere. A free website that hosts Twine Stories is philome.la. All it requires is a twitter account!
Here's an image that shows where the menu is!
I would suggest you to start a blog, its a very effective way to find readers, and do see: Tips for starting a blog.