The minor illusion spell uses the following rules in the last paragraph of the spell description for how a creature determines whether or not the illusion is actually an illusion:
If a creature uses its action to examine the sound or image, the creature can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
What is unclear to me is the meaning of "faint". English not being my first language I checked the dictionary and what I understand is that my object will be blurry, out-of-focus, looking fake… but it will still be there. On the other hand, my GM has ruled it is simply transparent for whoever looks at it (context was putting a 5-ft-by-5-ft black box in front of a window to block the view).
Does seeing through an illusion make it transparent?
If something is barely perceptible, you will only just be able to see it. At a glance, you could miss it. This would not be true if the illusion remained opaque, so it makes sense that it would be transparent once you discern it for what it is.
From a gamist perspective, your question notes one of the main uses of Minor Illusion - to block vision. If the object remained opaque, in a lot of cases succeeding on that check would provide no benefit to an enemy.
We can compare minor illusion to the very similarly worded 3rd-level major image:
A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the creature can see through the image, and its other sensory qualities become faint to the creature.
This spell uses the exact same mechanic as minor image for discerning the nature of the illusion, and it also describes the illusion as becoming faint after being discerned. It also explicitly says the image can be seen through.
These are two different spells, spells do only what they say they do, and just because one spell behaves in a particular way doesn't mean a similar spell also does. However, in this particular case it looks like the intent is that a minor image can be seen through once it has been discerned for what it is. Otherwise, the minor illusion cantrip would be more effective at vision-blocking illusions than a similar 3rd-level illusion spell, which would be an odd design.
Beyond that, we have nothing to go on besides the word "faint," which leaves it up to a DM ruling based on the definition of the word. (Other answers have covered this.) I've also avoided using "transparent" or "translucent" in this answer, since those can be subjectively interpreted. For mechanical purposes, it's sufficient that an illusion "can be seen through."