You can do that, just like with any other version control system. Visual Studio provides two ways to manage and version your source code,
1) Git (VSTS, Git for Visual Studio etc.)
2) TFVC (Team Foundation Version Control)
First you need to check what does your repository use, that will narrow down the search a bit. If your repository is using Git version control or some derived ones, then you would be using git to pull/push and commit the changes. On the other hand, if your repository uses TFVC then check in/check out is the way to go and you would need to have the setup to configure the TFVC on system — Visual Studio would solve most problems for you.
Quote:
"Open in Visual Studio"
It does open in Visual Studio, but you need to guide the IDE further to tell it to either clone the project, pull in a project or do what.
A version control is never as easy as, copy/paste or open the sln file. A version control is most likely to setup the repository on your machine and then track the changes. I recommend that you start off here,
VSTS and TFS Documentation | Microsoft Docs[
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Last year, I created a Playlist on YouTube for beginners, although it does not contain the TFVC in it but for Git based projects it can help you:
Introduction to Visual Studio Team Services - YouTube[
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