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Why should anyone need a French proxy to access a specific web site?

Well I'll tell you, all across the internet there are web sites tailoring their content to specific countries.  This is a concept called geotargeting, which in effect checks what country you're connecting from then customizes what you see depending on your location.
It actually sounds quite good, geotargeting and in some instance it is, for example when you go to a Google search page you always get directed to your own local version of the search engine.  Unfortunately this technology is just as often used to block access to content on a website.

For instance, one of the most famous sites on the web, the BBC Iplayer is restricted if you try to access from a Non-UK location, the big American media sites do the same restricting access to people outside the US for Hulu, NBC and ABC for instance.  Just about every major media site restricts it's content to those who reside in that particular country.

So for French TV you need to use a French proxy?
I'm afraid you do for the biggest sites, for instance M6 Replay will block the vast majority of their content unless you connect from a French IP address.  Of course this is the difficult part, proxies cost a lot to run especially streaming video down them so a free one is difficult to find particularly if you need it to be from a specific country.

So Is it Possible to Find a Free French proxy?
It is in theory, in practice it is an extremely time consuming task which you have to repeat everytime you want to access a site.  The difficulty is that most of the free proxy servers are just misconfigured or servers which have been hacked, they will only be accessible for a few hours until you have to find a new one.   You can use programs like foxy proxy to find them, but it will be an ongoing task especially if you need a French proxy.

Any other options for watching French media sites ?
You can configure a free program called TOR to route through French clients, which works ok as long as you don't want to stream video as the clients are much too slow to produce a watchable stream.

The only other route is to pay a provider for access to a private French Proxy server.  To be honest if you want to watch French TV regularly or access any of these sites, it's the only real option.  There are many companies who offer this service for UK and US sites but one or two that have French Proxy servers as well.
Have a look around there are a few providers, you need the connection either via a proxy or a VPN (Virtual private Network) and make sure they have a French server.  Some of the online companies make you pay for access to each specific server, however there are a few companies who allow use of their entire range of servers including french ones for a single subscription - the advantage is you can select which server you will need for each site.