The Aegon plot twist was set up in the House of the Undying in ACoK (the cloth dragon on poles, which Dany calls a mummer's dragon), and the first fan theories about Aegon VI appeared shortly after ACoK was published in 1998, so it's not completely out of nowhere. It's gained credence after increasingly common comments about Aegon's body being unrecognisable in ASoS and AFFC.

However, I think it's more likely he's a fake. The references to a mummer's dragon and in particular Daenerys being called the 'slayer of lies' in the same vision suggests she will expose Aegon as a fake.

As for how this was supposed to pan out, Daenerys' alliance with Khal Drogo was supposed to succeed and Viserys and the Dothraki would join the Golden Company before launching the invasion with the aid of Dorne. My guess is that Illyrio and Varys were letting Viserys be a very visible threat that would distract Robert (and this does at least explain why Varys seemed okay with the assassination attempt on Daenerys, since her role in all of this was minimal, just to win over the Dothraki; once she was killed, that could even be more useful in inspiring Drogo to attack Westeros). Once Illyrio became acquainted with Viserys and realised he was a fruit-loop nutcase, he probably reasoned he'd get himself killed by the Dothraki. If by some miracle he made it back alive and objected to Aegon, a little accident could be arranged for him.

Ultimately Aegon's revelation explains as why Varys and Illyrio's plans in AGoT seemed to revolve around either Dany or Viserys being expendable, which never made much sense if they are the last Targaryen claimants.

If Aegon is a fake, obviously he does have some Targaryen blood due to his looks. In that case I think he's either a descendant of a bastard born to a Targaryen (probably Aerion Brightflame, who is mentioned in ADWD as being a member of the Second Sons for a while) or a descendant of the Blackfyres in the female line; it is noted at one point that the male line was wiped out with Maelys but that suggests the female line survives.