By default WinSCP transfers files on foreground. This way it can use one connection to the server only all the time, however during the transfer you cannot use WinSCP for any other task. Alternative way is to use background transfers.
To instruct WinSCP to perform the transfer on background, check Transfer on background (add to transfer queue) on transfer options dialog before the transfer. If you do not use the dialog or want to transfer file on background by default, check Transfer on background by default on Background page of Preferences dialog. You can also move ongoing foreground transfer to background using Move to Background command. Also, you can drop files onto background transfers queue list to transfer them on background.
Once you instruct WinSCP to perform the transfer on background, the transfer is added at the end of transfer queue (visualized by background transfers queue list).
WinSCP by default performs at most two background transfers at once. If you schedule more transfers, those will wait in the queue until some previous transfer is finished. For each background transfer WinSCP opens separate connection to the server using the same session option as used for the main session. Once the transfer finishes the connection is kept on background and reused for future transfers.
Once the background transfer finishes, the target directory is reloaded (still on background) and content of the respective file panel is refreshed.
Once you have your transfers scheduled, you can use queue list to manage them.