As ilkkachu commented, the program whose output you redirected to a log file was designed to write status formatted for a text-console display, not for logging into a file.
The ^[[1m is an example of an ANSI control-code sequence for setting display attributes (foreground or background colour, bold, underlined etc). The ^[ is a typical representtation of an ASCII control character, in this case Escape (Esc) which is used in the relevant ANSI standard as a control-sequence introducer (CSI).
So this behaviour is baked into whatever program you are using. You might try misleading that program about what sort of output device it is connected to and thus fool it into not emitting those codes. Many programs use a library known as curses which uses the value of the environment variable TERM
The common shells allow you to temporarily change the value of an environment variable for a single command without affecting subsequent commands:
TERM=dumb programname options
^[[1msets bold etc. Whatever prints your log, probably prints them too, even though it doesn't usually make much sense in a file.