I'll try to be as convincing as possible without much exact textual back up. I do not have the book in front of me, but I think I can give you enough to help you understand.
The answer starts several chapters back - in the Mind Game. Remember when Ender first sees Peter's face in the mirror, and it is an updated picture? Graff (in the pre-chapter dialogue) is discussing (with Major Imbu I believe) the fact that the Mind Game has taken Ender to a place they've never seen before - The End of the World. He also asks how the Mind Game got a recent picture of Peter. Major Imbu cannot provide any exact answers, except that the Mind Game operates on its own level - beyond any human control.
Ender actually finds the bugger egg at "The End of the World" in the Mind Game first. What...
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(I believe) was really happening all along was that the Buggers were communicating with Enderthrough the Mind Game. One of Ender's strengths throughout the novel is his gift of empathy. He considers it a weakness every time he feels sorry for his enemy. But this is ultimately how he was able to save the human race and the Buggers. They tapped into his empathetic nature, communicated with him through the Mind Game, and essentially lead him to an egg holding a queen. The book ends on a note of hope - as Ender plans to take this egg to a new planet, let it live, and become the "Speaker for the Dead" - someone who will learn and tell the buggers story.
I've discussed the end of the book at length with the classes I've read this with - and certainly, it is ambiguous at best. I'm not sure that my synopsis is 100% accurate - but through several discussions I think this is what I've come to believing happened.
As for the other books in the series - if you liked the Ender's Game characters - you should definitely read Ender's Shadow, Shadow of the Hegemon, and Shadow Puppets. I didn't make it through Speaker for the Dead, etc. as it takes a much more science fiction turn.