How do ancestry websites work? Is it even possible to link back generations like that?

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Ancestry sites vary greatly in what they offer. None does the work for you. Common things they provide include:
-images of original records
-transcriptions of original records
-databases and indexes to help locate original records
-historical or geographic information
-tutorials
-catalogs of resources or lists of links
-forums for making social connections
-user-submitted trees

Ancestry.com may seem as though it does the work for you because of the leaf thing, but this is really nothing more than a sophisticated search engine giving you possible matches they have on file. User-submitted trees are about as reliable as Facebook. It's just a place to share your own work, and the quality of each person's work varies.

Can you link generations back? YES. That's what genealogists do using records that identify people and their relationships such as birth, marriage, death, census, and court records, etc. The basic process is:
1. Interview your living relatives.
2. Examine your documents and those of relatives that will allow it.
3. Prepare for research by learning about basic genealogy, genealogy specific to your known ancestors.
4. Organize your data. Free & low cost software is available.
5. Research one document at time for one generation at a time for one person at a time.

Ancestry/genealogy websites are just ONE of the tools available to help accomplish steps 3-5.