I have a different take:
Learning a new language allows you to see new ways of doing things (assuming you learn some of the ins and outs). Every language designed was made because someone couldn't do what they wanted in an existing language or the world it "lives in" (VM for interpreted languages, base library / stdlib for compiled-to-bytecode ones). Ok, so every serious language (leaving out the ones which are obviously just there because some people have a sense of humor (eg Shakespeare) and some people have a twisted sense of humor (Perl... ehm, I mean, whitespace!, yes, whitespace! And brainfuck! (: ).
Sometimes learning a new language makes me appreciate the features I already know in other languages (I'm looking at you, R), sometimes learning a new language teaches you new respect for formatting and "just getting along" (Python). Sometimes a new language teaches you about constraints (Lua, particularly embedded), sometimes about being mad careful about memory (C/C++), sometimes it's about what can be cobbled together with existing pieces (ie, quick wins) -- bash / zsh / sh / batch / PowerShell.
Sometimes it's about understanding the lowest-level stuff (asm, IL), sometimes about dealing in sets instead of procedures (SQL, R). Sometimes it's a reminder that more words may initially seem more readable, but just become a PITA (VB6 / VB.net -- and they are well different) or that even if a language seems like it's "for dummies", you can still accomplish the same thing as more "respected" languages (VB6 / VB.net again).
Sometimes learning a language makes you learn how to listen very carefully to its proponents, eg Tcl: "everything is a list" and "everything is a command" -- when that truly sinks in, there's a white-hot moment of clarity; it takes a day or two, but when it properly sinks in...boom. After that lesson, I give a little more thought to seemingly esoteric observations about a language.
Sometimes a language teaches you about the UNIX philosophy -- doing one thing and doing it well, then chaining together with other utilities which also do one thing well (F#, shell / batch languages) -- which shaped how I wrote "more capable" languages, producing smaller, simpler units instead of monolithic ones.
Sometimes a language seems more long-winded than it actually is and pleasantly surprises you (Java, particularly v7 on Android had some nifty tricks up its sleeve). Sometimes hacking something in a language gives you enough chills to not want to go back there (for me: Erlang, though I'm open to trying again) -- and even that is a positive learning experience.
Sometimes learning a language is just a lesson in history, of how things were once done and how, despite the forces against them, people can leave a mark on technology that is visible from space. I'm looking at you, Grace Hopper, and your creation: COBOL. There's a language structured around making your code look like something you'd find in a ring binder somewhere. And much of the world's financial institutions ran on that for, well, decades.
There's not one of them I'd wish to unlearn to get the time back. Indeed, I'm trying to get together the self-motivation to learn either Rust or Go, perhaps Haskell or Scala. Perhaps even Perl 6 (Larry promises that it's all better now). No particular reason other than perhaps it teaches me how to use my current tools better. Or perhaps I find another language which suits particular problems very well.
And yes, apart from whitespace and brainfuck (and, to be brutally honest, Erlang, which I hacked a little, but really, it doesn't count), which I've only read about, I've written code in all of the above. Some languages have been home for a long time -- literally tens of thousands of lines. Some I've dabbled in and moved on when I found something more suited to my immediate needs.
By "dabble", I mean only a few hundred lines or perhaps enough to understand how to port away from it (Perl, which never really managed to gain my approval, even though it's powerful stuff, but Perl seems to be a training ground for how to write line noise). But all have taught me something; all have been part of shaping my code as it dribbles out in the present, even if that lesson is how not to do it.
If you're a developer and you only know one language, you're doing yourself (and your team-mates and employer) a great disservice. Go out there and learn one of the wonderful, free languages that are available. If you don't know where to start, then pick the first one from this list that you don't already know: