Question: My internship mission consists in recoding an entire software which was coded 8 years ago by a current senior developer of the company in another language.

The only trouble is that I changed the organization of the software from bottom to top. I switched to MVC instead of the "all in a file" pattern for example.

How do I show him my current work without sounding like "your previous work sucked so much, that's why mine is so different" ?

Answer: 
When discussing your changes, avoid any use of "your old way" or "your old code". Once you join the development team, think of it as becoming "our code". 

Every developer knows that, given a few years, a new framework, development practice etc. could come along to do the work of a piece of code cleaner and faster. Whether it comes from an intern or a grizzled programming veteran, your senior developer colleague should have an open mind to examine your work fairly, so long as you are not sounding as if you are singling him/her out, or refusing to acknowledge criticism of your own work.
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> The only trouble is that I changed the organization of the software from bottom to top.
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Your other problem is apparently that you didn't discuss or obtain permission to do what you did, before you did it: because he's not expecting how much you've changed.

If I were senior developer reviewing a rewrite, one of my worries might be, "How can I tell whether the rewrite broke existing functionality?"

Do you have an answer to that question, e.g. are there defined system tests or acceptance tests, which the new version of the software is still passing?

If so then it's easier to review the new version on its merits, based on its superficial appearance and structure: e.g. to assess whether it's more modern, more maintainable, etc.

If not then you need to determine via code-inspection whether functionality has changed; which might be difficult if it's a complete rewrite.