Frankish kings treated their kingdoms as private property and the Frankish inheritance system had no conception of primogeniture (the right of the eldest son to inherit everything and thus keep the state intact), so they divided their kingdoms among all their sons – we can look at Lothair himself as an example, who later divided his kingdom between his three sons.

When two of Charlemagne's three sons predeceased him, his remaining son, Louis (later Louis the Pious), inherited the entire empire (with his brothers dead he succeeded uncontested). He quickly moved to secure the succession that would follow his own death and in 817 issued Ordinatio Imperii (the »Imperial Ordinance«), dividing his empire among his three sons. The eldest, Lothair was crowned as co-emperor and his brothers were crowned kings so that they may hold sway with regal power under their eldest brother. Lothair received the traditional Frankish heartlands since he was the co-emperor (and was to succeed as emperor in his own right) and in 822 Italy was added to Lothair's portion after the death of Louis's nephew Bernard, who had been king of Italy.

As to why Pepin got the western part and Louis (the German) the eastern I'm afraid I can't really say. If I had to speculate, I'd say that since Aquitane was Louis's (the Pious) own subkingdom while Charlemagne was still emperor and Pepin was older than Louis (the German), Louis (the Pious) felt that maybe his older son should »follow his footsteps« so to speak. But like I said, this is just guesswork.

Anyway, I hope this helps!