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Honestly I don't think a few paragraphs can do it justice. The term isn't even very well-defined - are first class functions enough? No side effects / immutability?

Personally I feel that while writing functional style code - by which I mean using passing functions as parameters and avoiding side effects - that can be help in most languages. But it offers a lot of guarantees that are really powerful if you can actually rely on them, which you can only do (imho) if a compiler verifies them. So I'm the die-hard that Jason Knight talks about, I guess.

Anyway, I would recommend to read a somewhat longer description of it. I am not very qualified to explain, and this answer would get way too long if I were. But a summary:

• Easier to reason about because no side effects (e.g. you know a value hasn't been changed because it's all immutable, and you know setting X does not affect function F because it's not an argument)
• Easier to test (same)
• Easier to formally prove correct
• More declarative: what you want instead of how to do it
• Easy to parallellize because there are no side effect so different paths cannot affect eachother (in fact sometimes the hard part if not parallellizing too much) - and parallellism is getting more important
• Enables a lot of optimizations, such as caching function results (since the result can only depend on the arguments)
• Often easier to add functions to classes of objects (but harder to add types)
• Can (optionally) do things like lazy evaluation, because there are no side effects
• Often very high level

The main downside being that things that do use side effects can get really complicated, like reading files. It's also far removed from how hardware works, which is sometimes a disadvantages but usually great.