What happened to Ichigo in Urahara's training area after meeting Byakuya (for the first time)? I understand that during his initial training with Urahara something occurred that gave him (or awoke in him) 


   hollow powers,


but hardly anything is said on the matter. And how come nothing like this never happened before, to the point of Urahara trying to make it possible. For Ichigo just happened?

And how does this play into


   Aizen's initial plan. Did he influence this event too?
It's pretty simple. When Urahara starts training him, in the end of chapter 60,


   Tessai breaks Ichigo's Chain of Fate. Urahara explains that the corruption of the chain has already begun, and when it reaches Ichigo's chest, he will become a hollow. The only way to stop that would be to regain his Shinigami powers. Then they throw Ichigo into the hole, and seal him with bakudou 99. 


In the end of chapter 62,


   the chain corruption gets to Ichigo's chest, and he starts transforming into a Hollow - you can clearly see his mask appearing (though Urahara comments that the mask forming before body indicates that Ichigo is resisting the transformation). However, Zangetsu helps Ichigo to stop the transformation in the middle and regain his Shinigami powers. 


So yes, the reason is simply that 


   he was already half-transformed into a Hollow when he gained his Shinigami powers back and stopped the transformation.