I thought that the native speakers spoke very quickly, without doing making any effort to speak distinctly.

I never spoke during the first few meetings at the beginning ["never" and "beginning" don't really match, as "never" is for all time and "at the beginning" is one point in time], because as, in the time I needed to understand what has been was said and think of what I can add, the conversation had changed of subject to a new topic! It was so frustrating.

But I stay and continued to come to the meetings. Several months later, I was able to follow what non-native speakers were saying, because they spoke more slowly slower and with the same accent as me, but I was still saying very little. so few words. [When you speak, you don't usually say a word here, a word there, which is what it sounds like you mean when you say "so few words"]

And, One day, there was something that was unlocked in me. I started to participate actively and spoke several sentences. I felt very happy, as because [you overuse "because"] I had reached the point where I was able to understand what was being said and to reply with new ideas. [I think the "imperfect" tense of "I felt" is what makes me want to say "what was being said"]

My friends were so surprised that I wasn't that baddidn't spoke so bad.

[This feels like "My friends were so surprised, that they said nothing for 5 minutes". Without "so", it is more neutral] 

They were pleased, too.
I noticed that I did made many lot of mistakes and that I didn't use the correct words, but it didn't matter to me. The meaning of what I wanted to say was quite well-understood, which This is the most important is what really matters [Even if you said "which is the most important thing", you'd need to somehow find a way to limit it to this setting, i.e. mention again that you mean in communicating with others. So, I made it a non-absolute statement by making it more colloquial/casual]. And, often My mistakes were the ocasions of often funny. moments and sometimes of laughting ones. [We don't say a "laughing moment" and usually don't bother distinguishing between when something's funny and when it causes someone to laugh. That is, I wouldn't say both, I would say one or the other]

Surely, finding [maybe you mean "creating"?] Italki communities [I don't know what you mean by "Italki communities". There are language partners, but no "community" option here] give me, and you, the possibility to follow a similar method of language learning I used. And, . With the help of Skype, we can easily create a small group of people that meet together once a month, once a week. Why not?
[I thought skype had an upper limit on the number of people in a group call?]

Only one rule : speak about everything but only in English for the first part then in French in the second part. [This doesn't quite work. You need an increment of time, not just "part", and it is effectively 2 rules, because you're talking about two halves of a converstion. Maybe something like "agree to speak for an hour or two and spend the first half speaking only English and the second half speaking only French"].  Please, contact me if this idea pleases appeals to you. [We don't really say "If it pleases you". It sounds either old fashioned or weirdly obsequious,]
Bye. [As this is not a letter, really, a closing is not necessary. Even if it were, "Bye" doesn't really make sense. I would say "I hope to hear from you soon!" or something like that.]

[1/2] Organising a regular meeting to improve a my language skills

I'm a French man, of nearly 50 years old.

I seldom travel a little  [this is subtle. If you say "I travel a little", i expect that the next part will say how travel is relevant for something. What you mean is that you do not travel much, therefore you don't have occasional to practice other languages. Thus, to make it more "negative", either say "I seldom travel" or something similar]

so I've rarely have occasion to practice other languages.

["I've" and "occasion" are different levels of formality. Contractions are casual, "I rarely have occasion to..." is formal. I've chosen to make your sentence more formal]

But [Technically, one shouldn't start a sentence with "But", much like "And"]  However, I always think ["always think" implies that it's something that enters your thoughts very very often; usually only used in reaction to a situation. E.g. "When he does that, I always think "he is acting like a child!""]

that to be opened to other cultures is very important, so I would like ["want", like "can" is too direct/crass/casual, and not really right for talking about goals/something insubstantial] want to be more and more fluent in English and Spanish languages. 

I think it is almost was about ten years ago [I know exactly the French formulation you are copying. It doesn't work in English (and it confused me in French!). The closest we have is "It has been almost ten years", but it follows a statement. E.g. "When was the last time I saw him? It has been almost ten years".] when a friend of mine, knowing my desire to be more fluent in Spanish,

 languages [It doesn't make sense to say you want to be more fluent in languages. You could either specify which ones, or say to learn more languages]

invited me to join an informal group of friends that meet once a month in a café in the downtown of Nantes [To me, no towns in Europe really have a "downtown", usually an old town. Maybe it's because in the states, "downtown" is always a horrible cluster of highrises and commerce, not a thriving, vibrant center of life]. It was a Spanish group with half of this the people as native speakers.

There was only one rule. We cancould [needs to be past tense] speak about everythinganything [again, subtle. I would never say I spoke about "everything" unless I said "everything having to do with..." or I want to convey that I spoke with someone for a long time, about every possible topic I could think of. "Anything" allows for, well, any topic] , but only in Spanish.

At the beginning, I found the conversations very hard to understand. [You use "very" too much]
Despite  In spite of I've been studying having studied Spanish language at school for 9 years and having lived being in Spain for 5 months, I quite understood nothing that was said.

["Despite" needs a noun, like "Despite his help";  

You never need to specify "language". "I am learning french" = "j'apprends le français", no need to specify that I mean the French language.

I also changed the tense, as, since you are 50, I assume you are no longer in school? "School" means pre-university;

I assume you are not currently in Spain, so I changed the tense]