Moba menus & configurations can be quite confusing and I too struggled to figure this out even after years of using this app, here's what went wrong:
It is most likely you saved previous machine session settings, while accidentally changed the default global SSH settings of your Moba client - let's fix it.
- Navigate to settings menu and select configure:

Select the SSH tab, and make sure to enable graphical SSH-browser and Automatically switch to SSH-browser tab after login:

Relaunch your MobaXterm app, and connect to your remote host - the file browser should open automatically upon successful login.