Why can we say that VI-VII prepares i in D natural minor ? VI is the chord B♭-D-F and VII the chord C-E-G. I am struggling to answer this question...
Firstly, D natural minor isn't a key. D minor is. And VI>VII will be the same in any minor key, not just Dm.
However, we'll speak in key Dm. VI is B♭ maj., and VII is Cmaj. And going to Dm after that is sort of expected. The direction is good. Actually, it's three quarters of the Andalucian sequence, backwards. Given that, if using only the D nat. min. notes, there's no leading note in the 'dominant' chord, that VII does act rather like the dominant chord in this situation.
It could also be said that CEG represents the top three notes of Am7, which tries to be the 'dominant' in that key.
VI (or vi in major) can function as a predominant, and VII can function as a dominant. And as with your prior question, this is not limited to the key of D, but is true for all 24 keys.
Thus VI–VII–i is a variation of the textbook PD–D–T (predominant, dominant, tonic) chord progression.
It's less common in the classical style, where the V chord reigned supreme. But in popular music of the last century or so, this progression is very common.
It can also happen in major (using what we call mode mixture), like in Billy Joel's "She's Got a Way." Here the V resolves deceptively to ♭VI, which then moves ahead into a ♭VI–♭VII–I progression.
I'll expand a bit on Tim's answer. The pattern VI-VII-i in any minor key is just three notes along the scale, ending on the tonic. Scale passages sound good however long (if not so long as to be boring.)
In many cases, composers in either major or minor keys treat the vii0 chord as a dominant seventh with an omitted root. (The vii07 is also treated as a dominant ninth without the root.) (There are cases were vii0 is treated as having step 7 as the root.) In the OP, one could by analogy treat VI-VII-i as an IV7-v7-i pattern which looks more like a cadence. The replacement of the upper version of step 7 with the lower (C♯ with C in this case) makes for a "softer" sound (at least that's the name that's been used for about 1500 years). In "tonal" (as opposed to "modal" rather than to "atonal") styles, the half-step and tritone resolution make the cadence "harder" (again the elder terminology).
So the pattern is very close to a IV-V-i pattern without the roots explicitly expressed.
I'll try to give an additional opinion:
D-minor is the relative key of F-major As we are in d-natural minor we have exactly the identical tone reservoir like F-major. The root tones of the scales are d (minor) and F (major). In F the perfect cadence (IV-V-I) is Bb-C-F, while there is also the false cadence Bb-C-Dm (IV-V-vi) which is VI-VII-i in d- natural minor. So you can see, the CEG chord that we hear as the dominant chord in F can also have a dominant function in d-minor (analogue to it's function in the progression to its relative key D major: bVI-bVII-I.