The flow elements are basically a fancy type of word wrap. See this page for some examples of how it is meant to function. The error you received in the validator is because the current version of SVG (and of that document) is SVG version 1.1 and flow elements are not part of the SVG 1.1 specification, so it causes the error with the validator.
The way flow elements are added to a document in Inkscape are:
Through the Text menu, via the Flow into Frame option.
By having the text tool selected, clicking and dragging the tool before entering text. Instead you should just click with the text tool instead of dragging it.
To remove the flow elements:
In your case the flow elements weren't being used, so I was able to open the SVG file with a text editor and delete the flow elements.
You can also remove flow elements in Inkscape via the Edit menu XML Editor, find the flow elements and use the del key to delete them.
If the flow elements were being used, you could either:
Manually edit the SVG with a text editor to remove them.
Open the file in Inkscape, find the text that is attached to the flow element and go to the Text menu and select Unflow.
If you're not worried about the text attached to the flow element you can also remove flow elements in Inkscape via the Edit menu XML Editor, find the flow elements and use the del key to delete them.
The SVG Check Tool does some basic checks to look for any elements that may cause problems. It will give you a warning about flow elements, text on path and some other issues that could cause rendering problems. The W3C Validator is used to make sure the SVG markup is valid with the SVG specification. If it isn't valid, it could lead to potential rendering issues which is why I checked the file with the validator.