Elementary OS 5 comes with a very limited systray functionality that needs to be fixed. For this you need to install wingpanel-indicator-ayatana that can be installed via a PPA and change autostart config of the indicator app.

Install wingpanel-indicator-ayatana

sudo add-apt-repository ppa:yunnxx/elementary
sudo apt update
sudo apt install indicator-application wingpanel-indicator-ayatana

Change autostart config

Open the config file with

sudo vim /etc/xdg/autostart/indicator-application.desktop

and change the line

OnlyShowIn=Unity;GNOME;

to

OnlyShowIn=Unity;GNOME;Pantheon;

Then reboot your machine and you should have a systray that displays your application icons too.