According to the Cambridge Dictionary, the verb believe is normally followed by a that- clause. The clause explains what you believe. According to this approach, the correct form would be:
I still can't believe that I got away with attending philosophy classes only 3 times through the year.
In informal speech, the that is omitted:
I still can't believe I got away with attending philosophy classes only 3 times through the year.
When you have a sentence where the clause expresses the degree or extent of something, it is normal to use how:
I still can't believe how stupid I was - adjective: degree of stupidity
I still can't believe how badly he reacted - adverb: degree of rudeness
I can't believe how many pairs of shoes she has - determiner: degree- count
This construction is sometimes used inappropriately when the that-clause is a simple fact, not a matter of degree, as in your sentence, where "I got away with..." is a simple fact:
I still can't believe how I got away with attending philosophy classes only 3 times through the year.
The use of how would be justified if you were to add easily, which would make it a matter of degree:
I still can't believe how easily I got away with attending philosophy classes only 3 times through the year.
This NGram, shows how very uncommon believe how is in writing, and if you look at the actual references you will find that most of them are not relevant- "I want to believe: how can I.." or express degree
I couldn't believe how much money I was making
I can't believe how blatantly Guy Ritchie ignored this in his film