You will access the server the same way as you would normally via RDP, the only difference is you can't walk over to your server closet or data center and console into it. You can set up file shares and mimic your on premises environment so it looks like nothing has changed to the end user.
Your server will now reside in your cloud providers data center as virtual machine: https://azure.microsoft.com/en-us/overview/what-is-a-virtual-machine/
Essentially you are renting space in Microsoft or Amazons data center and running your infrastructure there, IaaS (Infrastructure as a Service). You will have complete control of your server down to the kernel level, below that will be managed by your cloud service provider.
How to get started below:
First, sign up for a free trial account. Here is the link for Microsoft Azure: https://azure.microsoft.com/en-us/free/
Then, you can easily deploy a Windows virtual machine by following this tutorial: https://docs.microsoft.com/en-us/azure/virtual-machines/windows/quick-create-portal