Just now I saw the number of the badges go down from 13 to 5 then back to 13 and for now it shows 6.
Why is this happening?
Also, along with that, the number of people reached and votes cast is also reduced.
Short answer: The main site and meta are two different sites, each of them counts badges separately.
One guess how this can be explained is that you sometimes look at your profile on the main site where you have 13 badges (at the moment). And if you are on meta, you see the 6 badges earned on meta.
And, naturally, if you go on some other site (such as Stack Overflow or its meta) - you'll see the number of badges that you've earned on that particular site.