Learn French: Nous n'avons qu'une machine, c'est pourquoi nous allons la réparer. - We only have one machine, that is why we are going to repair it.

Both express a consequence, but c'est pourquoi (that is why) is slightly more formal and emphatic. It explicitly points back to the reason just stated (having only one machine) to clearly justify the consequence (repairing it).

Could I use par conséquent in this sentence instead?

Yes! Par conséquent (consequently / as a result) would work perfectly here. Both c'est pourquoi and par conséquent are great consequence connectors that elevate your French beyond just using basic words like donc or alors.

Why is there a n' before avons if the sentence isn't negative?

This is the ne ... que structure, which means "only" (literally "nothing but"). Nous n'avons qu'une machine means "We only have one machine." You could also say Nous avons seulement une machine, but ne ... que is a very common and natural alternative.

Why is the pronoun la placed before réparer and not before allons?