This dialog allows to reconcile your account with bank statement. Account balancing (or Reconciliation) is based on transaction lifecycle which is defined by transaction states. Every transaction has three states (statuses): uncleared, cleared and reconciled. The first status basically means that the transaction has been entered manually and you don't have any confirmation of it from the bank. One the bank reports it at the web site or in the statement, the transaction can be moved to the next state: Cleared. Once you reconcile all cleared transactions with the bank statement, all of them should be moved to the Reconciled state. The reconciled balance should be equal to the last statement balance.
Balance account dialog shows only cleared and uncleared transactions. The idea is to mark all transactions listed in the bank statement as cleared (checked). The Cleared balance at the bottom of the dialog must be equal to the current statement balance. Once you've identified and marked the statement transactions as cleared, just click OK button and all of them would be automatically moved to the Reconciled state.
Checking the record in the table toggles the Cleared state of transaction. Also you can use a Space key to do the same thing. Double clicking at the selected transaction allows to modify transaction data, for example, change amount of transaction if the bank statement reports a different number.
Learn more transaction modification dialog: Transaction Dialog.