Hello Fiery.
LogMeIn is, as you say, a remote access program that can be used to control and modify various settings of a computer through a remote computer. Such programs are usually installed by tech experts when dealing with complete computer novices to save both time and energy. If your boyfriend installed it on your computer, then he can use it to control your Operating System. And controlling the Operating System requires access of files which means he can view your history and your files just as if he were operating your own computer physically.
If you trust him, then you can leave the program as it is. However, it would be better if installed when called for and then uninstalled later on so that you can discreetly operate your own computer. In the end, it is you who have to decide to keep the program or not.
Hope that helps.
-- Goku