The reason you need to really care about memory management in game development is because most of the consoles don't have virtual memory. Once you run out (and you don't have that much), your game stops working. It's very similar to embedded system / real-time programming in this way. So you need to plan out how much space everything will use in advance, basically. Lots of fixed-sized buffers and pool allocators.
And the reason you don't use garbage collection is because real-time games can't afford a GC pause.