As you know, to make into means to transform [someone] into [something]. When it is used reflexively, then the same person (or group of people) is both the subject and the object of the action: that is, they are the person both doing the transforming and being transformed.
When I do a google search for the phrase "make yourself into", most of the top results are for various image-manipulation programs that will transform your selfies (photographic self-portraits). "Make yourself into a Sanrio character," for example, lets you change a picture of yourself into some hybrid with Hello Kitty or something like that. You are both the person doing the image manipulation and the person whose image is being manipulated, so the reflexive yourself is used.
Going back to your question, we have this sentence:
...struggle is what makes classes into themselves.
What this means is struggle is how a group of people transform into a class. You could paraphrase this sentence by saying:
A class is formed when a group of people join together to fight the same struggle.
Because that is essentially how Marx is defining the word "class," it also means that without struggle, classes wouldn't exist.