There are 3 areas...
Trim is the final delivered size of the piece.
If you order an A3 flyer, then the trim is the dimension of the A3 (29.7 x 42.0cm)
Safety is the margin inside the trim.
The area between the trim and where the actual content (text) starts. In most instances you'll want to leave at least 6 or 7mm between your content and the edge of the page (trim) to ensure nothing gets chopped off during the trim.
Bleed is the area outside the Trim and is "extra".
Nothing which is important should be in the bleed area. The bleed areas should only encompass extra bits of anything which you want to print to right up to the edge of the page (trim).
When trimmed, the paper can shift slightly. It's typically very accurate, but not always the same from job to job. It's best to not plan on a trim hitting some exact spot. The final dimensions will be correct, but the printed art may shift up/down/left/right slightly based upon where it's trimmed.
The Safety area compensates for a slightly shifted trim on the inside.
The Bleed compensates for a slightly shifted trim on the outside.
the new edges of the flyers
This is interesting.... "new" edges in what way? The edges of your flyer should not move. They are the Trim size. Any bleed should be extended outside the trim size. The edges should never be "new" really.
You do not mention what software you are using in your question. That may be the issue, or momentary hurdle.
When using software, in most instances, the Document size will be your Trim size. Then you add a 6-7mm inner margin and set a 3+mm outer bleed. Generally there's an area in the Document Setup to add a bleed amount...
It will look different based upon what software and what version you may be using. However, it's generally along the same type of thing.
The oddball here is if you are using raster editors such as Photoshop because Photoshop has no bleed feature in relation to the canvas like other software. You won't find a bleed area like the images above in Photoshop.
In such a case, you would increase the width and height of the canvas to 2x the bleed amount (1x top + 1 x bottom = 2x -- then the same for width). So if you have a 10x10cm document and need a 3mm bleed, you'd change the canvas to 10.6x10.6cm.
But you want all your content to be within the "safety" area.. so you should set guides inside the canvas to accommodate the 3mm bleed and a 6mm margin, although I'd use 7mm for the margin to make it easer and set a guide 10mm from each edge. Then all your important content needs to be within those guides to be safe.
Probably went overboard in explaining.
Note that I'm American. Therefore, using cm/mm is not normal usage for me. I did my best to use cm/mm above because that seemed appropriate based upon the question. I actually use imperial measurement ( .125" Bleed, .25" margin ) or preferably picas/points ( 1p6 bleed, 2 or 3p margin ).