I haven't found a way to see this easily in the UI. You can see the size of the individual snapshots by going to "AWS Backup" > "My Account" > "Backup Vaults". There, click on the vault you are interested in. Then click on the invididual recovery-points and their respective size is shown. Now add those values.
On the command line using awscli and jq, you can sum them for the whole vault (replace MY_XXX by your specific values):
aws backup --region MY_REGION list-recovery-points-by-backup-vault --backup-vault-name MY_BACKUP_VAULT_NAME --query 'RecoveryPoints[*].BackupSizeInBytes' --output json | jq add