When a flyer is printed on a commercial press it is printed on an oversize sheet, of either SRA2 or SRA3, often with many different flyers on the same sheet.
Once printed that larger sheet is then trimmed down to size, to get your A5, A6 etc. flyer out of it.
It is therefore necessary to have extra colour and/or background printed on the sheet if your flyer is to have print right to the edge.
This is what we call 'bleed'.
That is why when supplying your artwork, you should supply as an oversized document, so A5 148x210mm should be supplied as 154x216mm, so allow for 3mm of bleed on all four edges. This is then trimmed off when finishing.
To allow for small movements of the paper in the trimming process, and a decent border, we generally recommend a 5mm 'safe area' as well, which is 5mm in from each trimmed side.
This makes a total of 8mm before you should place any text or pictures which you don't want trimming off.
Please see the diagram below for a pictoral representation.