What happens during a live SQL Server backup?

Full backup contains both the data and log. For data, it simply copies each page of the database into the backup, as is at the moment it reads the page. It then appends into the backup media all the 'relevant' log. This includes, at the very least, all the log between the LSN at the start of the backup operation and the LSN at the end of the backup operation. In reality there is more log usually, as it has to include all active transactions at the start of backup and log needed by replication. See Debunking a couple of myths around full database backups.

When the database is restored, all the data pages are copied out into the database files, then all the log pages are copied out into the log file(s). The database is inconsistent at this moment, since it contains data page images that may be out of sync with one another. But now a normal recovery is run. Since the log contains all the log during the backup, at the end of the recovery the database is consistent.