Tyrion proves time and time again throughout the series that he's able to sweet talk his way into just about anything, and this apparently includes convincing Dany's dragons not to turn him into a grilled chicken nugget. Is this because he's a member of the Targaryen family, who's known for being able to tame and ride dragons? So far, no one else has been able to do anything remotely close to what Tyrion is able to in the Meereen basement where Dany locks up her dragons (remember this guy?). By acknowledging the fact that the dragons — Rhaegal and Viserion — are incredibly smart and probably understand what he's saying, they allow Tyrion to remove their chains.
On top of the dragons's clear acceptance of Tyrion, it's mentioned that Tyrion is fascinated with dragons in general. In the scene with Rhaegal and Viserion, he tells a story about how he begged his uncle for a dragon on his Nameday as a child, and the books hint that Tyrion has had dragon dreams (which is something Targaryens frequently experience).
