I know it's only an expression, but it doesn't get you closer to the birds, a telecoverter is just a lens which fits between your camera lens and body and has the effect of magnifying the image your existing lens is already producing.
The closest focus distance isn't affected, it remains the same except that subjects will appear bigger from the same distance using the converter .... 1.4X bigger in fact.
One thing you do need to be aware of is that you will lose one 1 fstop - so your 70-200 f/2.8 will become a 98-280 f/4 when the TC is fitted.
You should also check compatibility - my Nikkor TC14E-II did not fit my Sigma 70-200 as the recess was too shallow so it would have fouled the lens. Probably best sticking to a Sigma TC, and the Kenko's fit as well I believe