They don't really explain the issue properly. The issue is not with serializing data and putting it in the database. That is actually fine, for certain specific kinds of data. Not every kind of data is something that needs to be searchable using a relational database model. The settings for a plugin, for example, are generally stored serialized.

The issue happens when you store URLs or local directory paths in serialized data. People who are migrating sites from one URL or server to another might try the naive approach of doing an SQL dump, doing a search/replace on that dump, and then importing the dump to the new site. When they do so, they'll find that serialized data which they modified will be gone, because they forgot to change the length values to match correctly.

The solution is to make smarter search-replace mechanisms, or better yet to simply not store URLs and directory paths in serialized data. The WordPress core tries to avoid this, though it has not always been successful. If you find a place in core where URLs and such are being stored in a serialized format, the core team considers that a *bug* and would appreciate a ticket and/or a patch to that effect.

Migrating sites in that "naive" way is something you shouldn't be doing anyway, IMO, but that's a whole different argument... There's better ways to set up your dev/staging/production environments, without needing to change URLs around.