Why do CDs and DVDs fill up their data from the middle outwards?
Note the above assumption made in the question is incorrect.
For historical reasons writing (and reading) from the inside makes sense (different size disks are possible as explained in other answers).
For read performance reasons modern disks may be written (and read) outside in or even in both directions (dual layer).
Most disks are a standard size.
Commercially produced CDs and DVDs are not written but stamped on a press.
Although all DVDs (both homemade and commercial) utilize "pits" and "bumps" physically created (the pits on the unreadable side, and the bumps are on the readable side) on the discs to store the video and audio information, there is a difference on how the "pits" and "bumps" are created on commercial DVDs vs the way they are made on a home recorded DVD.
DVD movies you buy at the local video outlet are manufactured with a stamping process. This process is sort of like the way vinyl records are made - although the technology is obviously different (vinyl records are stamped with grooves versus DVD being stamped with pits and bumps).
Source The Difference Between Commercial and Home-recorded DVDs
The rest of this answer concentrates on the performance aspect.
The exceptions to writing (and reading) from the inside are XBox games (and other games consoles games) and Dual Layer DVDs (movies).
Xbox games have the data written from the outside in for performance reasons - since the outside is spinning faster than the inside the data can be read faster.
The Xbox games themselves are stored on DVD-9 (9GB single sided, double layer) formatted discs and are actually written in a very interesting fashion. The games are written from the outside of the DVD-9 discs to the inside, meaning that most discs will actually have the majority of their data stored around the outer perimeter of the disc.
Since the DVD drive in the Xbox is a Constant Angular Velocity (CAV) drive, it can read more data per second off of the outermost tracks making this a highly optimized way of storing data on the DVDs to reduce load times. Obviously the drive can also read regular CDs and DVDs as well.
Source Hardware Behind the Consoles - Part I: Microsoft's Xbox
Dual Layer DVDs (Movies)
Dual Layers DVDs can be written in either direction - there are two write modes.
Most movies are written as opposite track path. A movie will be split across across the layers, so no seek back to the inner edge is needed at the layer change.
There are two modes for dual layer orientation. With parallel track path (PTP), used on DVD-ROM, both layers start at the inside diameter (ID) and end at the outside diameter (OD) with the lead-out. With Opposite Track Path (OTP), used on DVD-Video, the lower layer starts at the ID and the upper layer starts at the OD, where the other layer ends, they share one lead-in and one lead-out.
Source DVD-R DL