The Standard Way
Android’s NDK now ships as an self extracting executable. You likely need to set the executable bit:
$ chmod +x android-ndk-r10c-linux-x86_64.bin
$ ./android-ndk-r10c-linux-x86_64.bin
The above will cause the NDK to extract into the current working directory.
Manual Extraction
Since the .bin file is really just a 7-Zip self extracting archive, you can manually extract the contents if needed:
$ 7za x -o/path/to/extract/to/ android-ndk-r10c-linux-x86_64.bin
7-Zip is available in Ubuntu for example via apt-get:
$ sudo apt-get install p7zip-full
Update
As of at least r14b on the NDK download page, we’re back to standard ZIP archives.