The Mac's Finder is the same thing as the Windows Explorer. Insert your Flash drive and you will see it become available in the sidebar of a Finder window under "Devices." If it does not appear here then try going to the Finder menu, select Preferences, and then in the Sidebar section checkthe option to include "External disks" in the sidebar (you can also check the option in the General section to have disks show on the Desktop).
When this is done, you can then drag any file from any accessible location in the Finder to the disk device in the sidebar and have it copy to the disk. You can also select items (use the Command and Shift keys in selecting to select more than one item) and then press Command-C to "Copy" followed by going to the disk, opening it, and pressing Command-V to paste the items to the disk and effectively copy them that way.
When finished select the disk and press Command-E to eject it, which can also be done by right-clicking it and choosing Eject, dragging it to the Trash from the Desktop, or by clicking the little Eject button that appears when you hover your mouse over the drive in the sidebar.