I know he got through the blockade of Valinor via the Silmaril, and I read somewhere that Manwë knew he was coming to Valinor and many Valar didn't like him being in their territory even though he was there to plead on behalf of men and elves. One of them, Mandos, even wanted to kill Eärendil for breaking the ban.
Well Eärendil is the son of Tuor (a man) and Idril (an elf). Men were generally banned from entering Valinor for a couple of reasons; although one is very clear the other may be based on opinion.
His Man blood
The general main reason for Men being removed from the Undying Lands is because Eru and the Valar meant it to be so. The Gift of Men meant they were removed from Arda upon death and received eternal peace, immediately. Men would find Valinor eternally boring and essentially burn out too quickly. See these thoughts written by Tolkien himself:
But in Aman such a creature would be a fleeting thing, he most swift-passing of all beasts. For his whole life would last little more than one half-year, and while all other living creatures would seem to him hardly to change, but to remain steadfast in life and joy with hope of endless years undimmed, he would rise and pass — even as upon Earth the grass may rise in spring and wither ere winter.
There is also another opinion, that Elves and the Valar are jealous of the Gift of Men, and therefore treat them with some disdain:
But as the years grow long and Time wears, even the Valar will come to envy the gift of Ilúvatar to the race of Men, that of liberation from the physical world, and the inevitability of loss and sorrows that must come with this existence within Arda.
They are bound to the circles of the world, where Men are not. They don't feel them worthy of what little they have over Men, for their liberation is reserved to the end of the world. It must be a torturous thing to wait for your judgement, where there's this other seemingly "lesser" race that has immediate recompense, for all their faults. I think Mandos particularly felt strongly about this because he is the one who receives Elves when they pass out of the living world and into his halls.
His Elf blood
His mother is descended from exiled Noldor. While he may not have directly been involved, you can see why a Valar like Mandos would be particularly argumentative about a member of a family that carried out the first kin-slaying (some of which no doubt were his friends), coming back to his safe haven. He even set about the Doom of Mandos, which ultimately resulted the exile. You can't blame him for being a bit over cautious over the whole thing.
Why was he allowed in and why did they listen to him?
Manwë revered Eärendil for his bravery, and because of this sought mercy on him contrary to Mandos' pleas to have him killed for his impertinence. It's a clear theme in Tolkien's works that forgiveness and mercy should prevail over false retribution.
So there you have it. Really anyone opposed to Eärendil arriving had some sound reasons to be annoyed and apprehensive. But the boss on top of the big house said "It's ok, we'll give him the benefit of the doubt".
The answer is pretty simple:
Because he was mortal.
Remember that Valinor was also known as "The Undying Lands". It was a place where the Elves and Valar could live (being immortal) in peace, where everything was preserved and was eternal. In such a place, death was extremely rare, and was a cause of great grief and sadness. A mortal being in Valinor would inevitably bring death, grief and unhappiness there.
Secondly, he could not return to mortal lands and speak of Valinor, as in comparison Middle-earth/Beleriand at the end of the First Age was almost completely a warzone, and was really dangerous and full of death. If he spoke at all of the peace and splendour of Valinor to mortals, it would have triggered a mass exodus, and even if they were accepted to the Undying Lands, they would bring yet more death, grief and sadness there. See point 1.
As such, it was decided, for the sake of both races, that mortals should never set foot in Valinor, as they would be filled with desire for it, and never be at rest, meanwhile the Valar and Elves would have to see death again, and have to deal with all it's consequences and ramifications. Including the reminder that the former were all bound to the world until it's breaking, while Men could leave upon death.
Eärendil broke that ban (somewhat unwittingly), and thus Mandos (primarily) demanded his death. This was not punishment, but to protect what bliss was left in Valinor. Also it would have been better for Eärendil to have seen Valinor and then died, than to spend his remaining years in Middle-earth, yearning for the forbidden West, never knowing happiness.
If he had been pure Edain, I have no doubt he would have been taken to Mandos and killed, albeit humanely, but it was his half blood that saved him.