The A-Max cartridge is used for two things:
1) It has sockets for a set of Macintosh ROM chips (either 64KB or 128KB total size). The A-Max program reads the ROM data when it starts up.
2) It allows an 800KB Macintosh-compatible floppy drive to be connected, so you can use 400KB & 800KB Macintosh disks. Other programs such as CrossMac and ShapeShifter work with Mac disks via the A-Max cartridge and mfm.device.
The ROM-reading process is quite slow. There is a patch on Aminet to dump the ROM data to a file, and the patched A-Max executable reads that instead (and so no longer requires that an A-Max cart be present).
For what it's worth, the latest version of WinUAE can emulate the ROM-related part of the A-Max cartridge, which allows A-Max to be run unmodified under emulation.
