Q:

Why is Illyrio Mopatis willing to give so much to Viserys and Daenerys Targaryen?

In the first season of Game of Thrones, we see how much Illyrio is giving Viserys and Daenerys: shelter, food, and above all the priceless dragon eggs given to Dany.
Why is he doing it? What's in it for him?

A:

There probably isn't an in-tv-series instance to prove the following, but the books give better insight into the relationship between Illyrio Mopatis and Varys (Emphisis My Own)...

As a young man, Illyrio was a poor bravo, living by his blade. After
  the young Varys arrived in Pentos they formed an unlikely friendship
  and worked together and became very rich. Illyrio grew very
  respectable and married the maiden daughter of a cousin of the Prince
  of Pentos - only with the help of Varys did he reach a position
  where the daughter of a magister was willing to marry him. After
  she died, he remarried at least once, to Serra, a Lysene pillow house
  worker. This infuriated the prince, who barred the palace to him
  forever after. Illyrio was happy with Serra until the Treasure, a
  Braavosi trading galley carrying the plague, anchored in Pentos.
  Through the rats the plague left the ship and killed two thousand in
  the city, among them Serra. Illyrio keeps a locket with Serra's
  picture and her hands as a reminder of his lost love.
Even decades after Varys left to serve in Westeros the two still keep
  in close contact.
Illyrio has kept an eye on the remaining Targaryens, Viserys and
  Daenerys, for years whilst they lived in exile in the Free Cities.
  Eventually, he took them into his manse, keeping them under his
  protection. He was smitten at first by the beauty of Daenerys and
  thought of eliminating Viserys and claiming her as a wife, but
  dismissed this idea soon as he believed the girl too timid and
  frightened and that coupling with her would give him no pleasure. He
  claims he protected Viserys and Daenerys on the grounds that Viserys
  promised to make him the master of coin upon his ascension to the Iron
  Throne.

So in short, Illyrrio believed he owed something to Varys AND he then later also believed, that if he would follow through on protecting the Targaryen children, that Viserys would make him master of coin after Viserys became King of the Seven Kingdoms.

A:

There is almost nothing in the show about it, except that glimpse from Arya, but more is in the books. Illyrio have similar motives like Varys. They both come from poor background and actually care for greater good, and doing any deed necessary for that end. They make plans together, making sure they are completed on their respective side of the world. 
Perhaps Illyrio does have a feeling of owning something to Varys but their relationship is way over that. Also, he is incredibly rich, so it is not likely he is wasting time on Targaryens purely for personal gain. Also, he has no intention on living in Westeros either. And in the books, we discover that Varys and Illyrio have multiple plans and potential rulers of Westeros. 
To be clear, Illyrio did make profit in selling Targaryen bride to Dothrakies, but main purpose is having replacements for bad kings in Westeros. Which is why they did business with Khal Drogo, otherwise King Robert would probably send them all gold from Westeros for their heads. Personal gain is always there for both Varys and Illyrio, they have to stay powerful, but it is by no means their primary motive. Which is why we had a glimpse of them together below the Red Keep when Arya was chasing the cat. They both sensed the war is coming and couldn't plan their next moves by messages. They are not Targaryen supporters, they would surely replace Viserys once they realize he is cruel, like we saw change of Varys towards Dany in the show. So, keeping Viserys and Dany just means more cards to potentially play.

