Ahrefs subscription plans allow specific number of users (Lite and Standard plans - 1 user, Advanced and Agency - 3 and 5 users respectively). You can get more info on it here:

How do I add a new user to my account?

By users, we mean a number of simultaneous sessions, so if you exceed the limit, you might experience an issue with being kicked out of your Ahrefs account. In order to avoid it, make sure you terminated all other sessions before signing in again.
You will also see this message:

In some cases, however, you still can be logged out after trying to access some of the tools. You might see something similar to this:

If so, simply, clear cookies in your browser for ahrefs.com , and it should do the trick.
Let's take a quick look how it can be done in Chrome and Firefox browsers.
Chrome:

  1. Open ahrefs.com and right click somewhere on that page. 
  2. Then, choose Inspect at the bottom:

3. Once you have that go to Application tab and find Cookies in the first column:

4. Right click it and choose Clear.
Firefox:

  1. Open the menu by clicking on the respective icon in the top right corner and go to Options:

2. In Options, navigate to Privacy & Security tab and open Manage Data window:

3. Type in ahrefs.com in the search bar, click on the result that came up and hit Remove selected button. After that, save changes:

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