If you want to export a large number of search results from a database, you may be able to export a file from the database and import it into Zotero.

There are a few things to keep in mind:

Each database has a different process for exporting a set of search results. Refer to the help documentation for the database or contact your librarian for more information.
Most databases allow you to export the results in various file types. You will need to export a .ris file (or, in the case of PubMed, select "PubMed format" which will produce a .txt file that Zotero can interpret).
Some databases will ask you how much information to include from the export. Review this carefully because it may not automatically include information you want, like the abstract.

When you have your file ready, open the Zotero program and choose 'File' > 'Import.'

The Zotero program, displaying File > Import.

You will be prompted to specify that you're importing a file. Click 'Next.'

The Zotero program, showing import options.

Next, you'll be asked whether you want to put the imported items in a new folder. If you choose this option, a folder will be created that has the same name as the file. (You can rename it later if you wish.) Click 'Next.'

The Zotero program, showing import options.

You will be prompted to select the file from your computer. Once you've selected it, you should see all items appear in your Zotero program.

The Zotero program, showing imported references.