Is it really the fact that vast majority of web developers uses OSX? Is it true that as a web developer I am expected to use Mac OS?
No, but it
is a popular choice.
Websites and applications are most commonly hosted on Linux servers. MacOS is a Unix system so, under the hood, the file system, command line etc. are very similar to Linux, and nearly all the (free) tools and applications used for web development and servers under Linux are available for Mac. So you're developing in an environment similar to the one to which you will deploy. Install a package manager like "macports" or "homebrew" and you can install whatever versions of Apache, PHP, Node.js, MySQL, MariaDB, mongo, git, nginx you want to target (they're mostly available as standalone Mac packages too, but if you need a bunch of them and want to match & switch versions its worth using a package manager).
Of course, Linux would be even closer, but on the Mac you can also run many key "industry standard" PC applications like MS Office and Adobe Creative Suite(...even if these aren't your first choice, you're often forced to use them by clients or colleagues) as well as many excellent Mac applications (such as Pixelmator and Affinity Designer as cheaper alternatives to Adobe) which tend to be better than the functional but rather clunky alternatives on Linux.
There's no reason you can't do web development on a Windows PC - all the tools are there (and you can always run a local Linux server in a virtual machine - plus, give the new Windows Linux Subsystem a year or so to get out of Beta and things may change) but the environment is very different (different rules for pathnames, drive letters, Powershell vs. bash). Then, of course, you might be targeting Windows server technologies (IIS, SQL Server, .Net framework) in which case, using a PC and Visual Studio probably makes sense. Still, with a Mac you can always run Windows under a virtual machine.
If so, Is Macbook Air enough for that?
Yes. Unless you're also doing
serious graphics work or video editing (which sometimes go hand-in-hand with web development) an Air should be fine (It'll happily drive a 27" 1440 display).
Just go for 8GB RAM and at least a 256GB SSD (not the entry level 128GB): its likely that you
will want to run virtual machines with Windows (if only for testing sites on IE/Edge) or Linux (simulating servers) or install something like Docker.
Of course, spending more on a MacBook Pro
will get you a nicer, faster machine with a far better display and better long term prospects - but it will also nuke your finances!
But lets be realistic here - if you're on a shoestring here,
a PC will do the job and let you hone your HTML5, CSS, Javascript or whatever skills. At the end of the day, web development needs a text editor and a web browser, and any personal computer sold in the last decade will do. (1) Get good at HTML5/JavaScript - on a $35 Raspberry Pi if you have to - (2) Get good job (3) Profit (4) Buy Mac system of your dreams. In that order.
I planning to get a full-time job as a web developer soon.
If you've got prospective employers in mind, ask questions or look at job ads and see what they're asking for.