On selection of List title from the populated dropdown you need to make call to SharePoint to get the Item IDs. Now to handle the List selection event on the first dropdown you can use onPropertyPaneFieldChanged(propertyPath, oldValue, newValue)
method from the BaseWebPart Class
Reference
Microsoft Docs: onPropertyPaneFieldChanged
The implementation of onPropertyPaneFieldChanged will look like this
protected onPropertyPaneFieldChanged (propertyPath :string,oldValue:any,newValue:any ):void {
if (propertyPath == `selectedList`) { // property pane list dropdown name
//code to get the list items Id using the rest call
//bind the items with second dropdown
this .context.propertyPane.refresh (); //this is required }}
You can also refer: https://www.sharepointwidgets.com/2018/05/sharepoint-framework-spfx-cascade.html for similar implementation(List views are loaded dynamically based on the list selected)
OR
https://docs.microsoft.com/en-us/sharepoint/dev/spfx/web-parts/guidance/use-cascading-dropdowns-in-web-part-properties