This can be a challenging situation. Craft content can be highly relational which makes it tricky to just export and re-import.
Scenario 1: If your database and your clients database were both equivalent before your client added the additional content, you may be able to just migrate over your clients database and replace your own database and get both back in sync.
To migrate the database would involve exporting the database from the clients environment (either via Settings->Tools->Backup Database
or via a database admin tool) and reimporting it into your environment with a database admin tool.
Scenario 2: If both you and your client made changes to your respective databases (for example, you both added articles in each environment) and you need to preserve the new articles added in your environment while also migrating over the articles added in the clients environment you have two options.
First, you could just manually re-enter the articles in the right environment. As it sounds like your client added a lot of articles, that might be time consuming, but sometimes these types of mistakes take time to correct manually. It's often the best option.
In the case you added a few articles and your client added many articles, it may make more sense to add your articles to your clients database and then migrate the full database back to your servers as described in Scenario 1.
Your second option would be to migrate the content programmatically. This becomes a bit more technical so you'd probably need to get a front-end or back-end developer involved. And it will only make sense to go this route if the quantity of data you need to migrate is too much to just do manually.
A plugin like Sprout Import can help with this scenario. (Full disclosure, my team developed Sprout Import.) The steps would involve outputting the content from your clients database into a template into an importable JSON format. And then importing the content defined in the JSON file into your database. The docs and examples within the plugin would go into more detail about the exact format you would need. For this answer, the more general point is that if your data was complex enough, a database migration plugin like Sprout Import could help provide a path toward a solution.