In DOS you needed to load ANSI.SYS in your CONFIG.SYS file in order to see ANSI terminal codes. Later on PC Magazine or some other company made ANSI.COM that you could load at any time as a Terminate and Stay Resident program to see ANSI codes.

It basically uses the CGA 80x25 text 16 color screen on PC and PC clones.

I could be wrong, but I think the terminal codes were based on VT-100 or something with the IBM font added. When I had an Amiga 1000 I had to set my terminal to VT-100 and load a font that had the IBM PC character set in it, because the Amiga fonts had different character sets in them that didn't look right.

Modern terminal programs for BBSes are like this one: http://syncterm.bbsdev.net/

It fully support BBS ANSI codes. It also lets you connect to Internet based Telnet BBS systems.