I disagree with splishnsplash. Creating a silo is relatively easy, but you can't do it right from the dashboard. You're going to have to customize your theme by editing the raw PHP files.
Homepage links to each category. Each category has a couple of posts related to the category.
Posts in each category link to the other posts in that category, the category page, and the home page.
Posts do not link to posts in other categories. Posts do not link to other category pages. Categories do not link to other categories.
The end result is hubs of relevant information linking to each other, their parent category page, and the homepage. If you interlink within categories, then it's completely useless.
If you do it correctly, search engines will be able to easily tell which part of your website is about what. And all of your sites will pass around relevant juice... all of the silos will boost the index, which will pass down relevant juice to each category. The categories aren't related to each other, but the homepage is related to each category.
Let's take the animal example...
Homepage: animals
Categories: cats, dogs, and hamsters
Each category has 5 posts within it - dog training, dog food, dog breeds, etc. The page about "dogs" is just a general page with info about dogs.
Dogs do not link to cats in any way. Cats do not link to hamsters, etc.
HOWEVER. They ALL link to the homepage, because each post and category page is relevant to "animals". The homepage takes that juice, makes it relevant, and passes it down to the other categories. The other, completely irrelevant categories are now receiving relevant link juice because it's been cleaned up by the index.
The tough part is making sure that there is no duplicate content anywhere. Make sure you use custom excerpts if you're displaying them anywhere.
You basically can't use any WP plugins to display other posts in the category because I don't know of a single one that does this effectively. You CAN make your own "plugin" out of PHP in the sidebar to display other posts in the category on the sidebar and the category page that the post is on. Another option is interlinking within the article... this is the safest bet. Automatically inserting other posts from the category is asking for trouble. Also overoptimization and too many exact match anchors and all of that.
I haven't made a silo in quite a while, so I'm not sure if there are out-of-the-box solutions. Something tells me that there still isn't. Half of the battle is setting up a silo... it'll take you forever on your first one. There's a lot of maintenance that you need to do, particularly with interlinking the posts in the same category as you add new ones.
Oh, one more thing. You can't use many of the standard WP layouts. Each post can ONLY link to the other posts in the category, the category, and the index. Each category can ONLY link to the index and other posts in the category. This is where customizing the theme comes from.
You know what... now that I type this all out, I guess making a silo is pretty complicated. I actually agree with splishnsplash, but I have never typed this out. I'm not even sure if they're still effective. But the moral of the story is that YES, YOU CAN MAKE WORDPRESS INTO A SILO, you just have to know how to do it. (WordPress doesn't "leak" anything unless you CONFIGURE IT TO LEAK SOMETHING.)
Good luck!
Jared