It looks like there’s very little text on the home page, a lot of it is “baked” in images, so if you craft a query that produces the home page, search engines are forced to use the first text they encounter.
Search engines often will ignore the description (not sure about the criteria for using it), so that it isn’t crafted as self serving for the website, as opposed to as useful for the search engine user.
If you search for text in one of the pages containing an article with a reasonable amount of text, search engines will produce the proper snippet from the page, for example “Non sempre gli effetti speciali vengono utilizzati per creare elementi o visioni del fantastico”.
About the home page my suggestion would be to replace images containing titles with actual text. It might be a bit of work to find a style to match the images, but search engines will then find that. Additionally you could pull some text from the linked article or category description into the source page.
In practice it’s not really a problem if search queries don’t hit your home page, people don’t necessarily search for the name of your site, they are likely to hit some query term that matches one of the inner pages.
This is in fact a common mistake: focusing on a single term or phrase that in your mind is the summary of your website content. Rather you should think of several query terms/phrases.
Finally, using your own browser to monitor your rank in search engines is also not a good measure. Browsers often personalize the search results for you, based on your past searches and past clicks. Your location also can affect the search result ordering. It’s best to use a SEO tool that monitors keywords/keyphrases over time and tracks their change over time.