How to ask professor to let me audit lectures at university I am not enrolled in?
First I would check if the school has a formal procedure for auditing. For example, my school wants auditors to pay as if they were attending the class, and they also have insurance concerns if, e.g., a student is hurt in a chem lab and it turns out that they weren't enrolled. You can get this info from the school's web site or catalog.

If that's not an issue, then I would just send a brief email and ask permission. There are unlikely to be any objections, but it would be polite to ask. Some conceivable problems:


There might be a shortage of seating.
In certain subjects (e.g., languages) or in classes using certain teaching methods, it might be expected that students will participate actively. The quality of the discussion or activities might be worse if there is an auditor who isn't taking the course seriously.
The professor might want to talk to you for your benefit to see if you have enough of a background to get anything out of the class.


Personally it doesn't bother me if an auditor's attendance is irregular.