It was ... advised to me by the "student advisor" to let my thesis supervisor know about my problem.
However, I am not sure how to go about doing this in regard to how specific I should be in describing my issues.
I'd say you've done the right thing by speaking to somebody in an official capacity. That first conversation can be the hardest part! Assuming that this "student advisor" is a member of staff (in your department?) I would suggest speaking with them again, and discussing some of your concerns about "how specific" to be.
If, however, I am misunderstanding the term "student advisor", and the person you spoke with was instead somebody in a counselling/support role, I would suggest speaking with an appropriate member of academic or administrative staff -- in my experience, that would be a "Director of Studies" or similar, but the Dutch system may differ!
Either way, "the university" should be able to tell you what is an appropriate level of detail, especially if you have concerns about putting your supervisor into a conflicted position with regard to your grade.
When it comes down to actually talking to your supervisor, I would suggest that beforehand you set yourself a minimum level of detail that you feel is necessary to convey, and a maximum level that you feel comfortable with. That might help you feel a little more in control of the situation. As for what exactly you say, you can then go with whatever seems comfortable (for both parties) when you're having the actual conversation.
One possible option would be to let him know that I have some sort of mental health issue. The benefit of this would be that he might be somewhat more receptive and understanding of my decreased productivity (and possible other problems).
On the other hand I'm afraid it might be a bit harsh on him, because he might me reluctant to mark my thesis as insufficient or to have tons of negative commentary.
If I remember rightly, when I was in a similar situation to you my supervisor was wise enough to ask me what would be (un)helpful in terms of negative commentary etc. Your supervisor may also volunteer to have that discussion; however, if not, you can always bring up the topic -- if you fear that he'll be reluctant to give you (constructive) negative comments, you can say explicitly "I don't need you to be gentler with your comments; I just wanted to let you know that this is the situation, and that accordingly I am having some difficulties" (or whatever wording best addresses your concerns and desires!).
As a quick personal note (which, hopefully, helps anyone having doubts about talking to their university regarding mental health issues) I should say that after years (and several degrees) dealing with these kinds of problems, I've never regretted talking to a member of staff about my mental health; I do, however, regret the occasions where (I now know) I could have let somebody know that I was having problems, but didn't. It's very easy to be nervous about these conversations, but they're often the best way forward.
N.B. While I have been in a similar situation, I should note that in my case it was/is a PhD thesis (and is in Britain), so my supervisor is not the one responsible for final grading of my thesis. However, your concerns regarding what level of detail to go into sound very familiar to me, as does your worry about putting your supervisor in a difficult position with regard to his comments on your thesis. So, hopefully, this answer remains helpful.