When you have connected boards, there should be a way to click to jump to the connected board with a filter applied that only shows the entries that were connected.
For example, you have a Company board with a column connecting it to Contacts. You have 10 Contacts connected in that 1 column. You could be able to, from the Company board, click that column and jump to the Contacts board, with a filter applied automatically, that just shows those 10 contacts.
As someone heavily relying on connecting boards within the CRM product for contact and project management, I know there is a serious need for this feature.