The software writes into my databases or gather any information?

Absolutely NOT. For any reasons the software writes (insert/updates/delete) any information in your databases. The technology I use to interact with the servers and databases are SMO Objects which is a Microsoft SQL Server ® technology. I DON'T gather any information about your servers, databases or data.

What kind of user can make backups?

Usually you only need to be sure that the user you choose to configure the backup task has the grants to do a backup operations in your servers and databases. With the option REMOTE (Script) you have to be sure that the user has access to read the sys tables. Example 'sys.types' and 'sys.schemas'.

How works the REMOTE (Script) option?

This option instead of make a '.bak' file, generates a script file with all the database schema and data. Be aware that with the TIMESTAMPS data type the behavior is the same as if you generate your script with Microsoft SQL Server Management Studio, and this is simply not to generate the insert specific to this type of columns. You have to be sure that the user has access to read the sys tables. Example 'sys.types' and 'sys.schemas'.

What it is MS SQL Server Backup Compression optional?

Since MS SQL Server ® 2008, Microsoft introduces the capability of compress natively the backup. Not all the editions support this option. I suggest always check this feature, if it is not available doesn't matter, because SQLServerBooster internally always check it before apply. If you want to know more about this feature read this article in MSDN.