"It had a high memory footprint" is just another way to say their program used a lot of RAM. It is related to garbage collection in that GC'd programs only free memory periodically (because each GC cycle takes CPU time), whereas manual memory management tends to free memory more or less as soon as it's unused.
The downside of manual memory management is either that mistakes can cause crashes and security bugs (as in C++, where you can accidentally use a freed variable after the memory has been reused for something else) or you have to put effort into expressing the exact lifetimes of each variable, reference, etc. in your code so that the compiler can check that they're being used in a valid way (as in Rust, where you interact with the borrow checker to root out potentially incorrect uses of memory in your code).
The sentence in the Wired story makes it sound like "memory footprint" is a simple measurable quantity you could assign to any language (and your question takes that idea to its logical conclusion). It's not quite that simple. In different languages, doing different things has different costs in memory, performance, and so on, and you kind of have to understand languages'/runtimes' details to know how the language will work with a given sort of program.
For example, CPython has reference counting, and that frees unused memory sooner but at the cost of having to store and update reference counts. Java has, on the one hand, things like object headers that add a certain amount of memory overhead per object, but uses some tricks to speed garbage collection (like generational collection) that Go doesn't (yet). Or in Go, you might try to reduce the memory footprint of a program by recycling memory with free pools and adjusting GOGC to free unused memory more often as kostya said.
The bigger point there is not that those specific details I listed are super important, but that there can be a lot of details to consider other than "higher memory footprint" or "lower memory footprint."
So: "memory footprint" refers to the amount of RAM a particular program with a particular workload takes up. Bigger picture, it's one factor in a large set of tradeoffs that folks like you or I or Dropbox's team have to navigate.