A more complete historical look Cygwin is Geoffrey J. Noer's 1998 paper, "Cygwin32: A Free Win32 Porting Layer for UNIX® Applications" which can be found at the 2nd USENIX Windows NT Symposium Online Proceedings.
Cygwin began development in 1995 at Cygnus Solutions (now part of Red Hat Software). The first thing done was to enhance the development tools (gcc, gdb, gas, etc.) so that they could generate and interpret Win32 native object files. The next task was to port the tools to Win NT/9x. We could have done this by rewriting large portions of the source to work within the context of the Win32 API. But this would have meant spending a huge amount of time on each and every tool. Instead, we took a substantially different approach by writing a shared library (the Cygwin DLL) that adds the necessary UNIX-like functionality missing from the Win32 API (fork, spawn, signals, select, sockets, etc.). We call this new interface the Cygwin API. Once written, it was possible to build working Win32 tools using UNIX-hosted cross-compilers, linking against this library.
From this point, we pursued the goal of producing native tools capable of rebuilding themselves under Windows 9x and NT (this is often called self-hosting). Since neither OS ships with standard UNIX user tools (fileutils, textutils, bash, etc...), we had to get the GNU equivalents working with the Cygwin API. Most of these tools were previously only built natively so we had to modify their configure scripts to be compatible with cross-compilation. Other than the configuration changes, very few source-level changes had to be made. Running bash with the development tools and user tools in place, Windows 9x and NT look like a flavor of UNIX from the perspective of the GNU configure mechanism. Self hosting was achieved as of the beta 17.1 release in October 1996.
The entire Cygwin toolset was available as a monolithic install. In April 2000, the project announced a New Cygwin Net Release which provided the native Win32 program setup.exe to install and upgrade each package separately. Since then, the Cygwin DLL and setup.exe have seen continuous development.