Why does Windows not recognize files inside Linux partitions?

Solution 1:

Windows only natively supports the NTFS and FAT (several flavors) file systems (for hard drives/magnetic systems) and CDFS and UDF for optical media, per this article.

To access other file systems, additional drivers/software will be required. As an example, the Ext2 Installable File System For Windows driver is an open source project that supports the Ext2 system.

EDIT: Correct typo - listed "UFS" instead of "UDF" - kudos to @ChrisS for catching the mistake.

Solution 2:

Windows does not have native Linux filesystem support (ext3, ext4, zfs, among others). It's as simple as that.