떼다's most common meaning is to detach, or to take something off (from an objec, like wall for example, or a book, etc). It is also used in the sense of 발급받다. 발급받다(get a document or certificated issued to you) might be considered as similar to 떼다, because if you meet the requirements (prove your identity, submit the required documents, etc), they give it to you by going through a process similar to detaching a sheet from a pre-made book of blank certificates (i.e. 증명서를 떼어서 주다). It is like taking your own copy from a book or stack that is ready and waiting for you.

'출입국사실증명서를 발급받다/떼다' means, issue a certificate that proves your entry or departure of any country.