Longer answer: cardinal systems (the generalized form that approval is part of) is able to capture more information about voter preference. It is easiest to see this with something like range voting. You may be familiar with a question like "Do you agree or disagree with x statement?: strongly disagree, disagree, neutral, agree, strongly agree." That is range. Approval would be "Do you agree or disagree with x statement?: Disagree, agree". Conversely FPTP would present you many statements and you have to pick the one you agree with the most, even though you may agree with many in the list. You can probably see here that you are actually able to capture a fair amount of preference information through this, thus allowing to elect representatives that more accurately represent the population as a whole.
Longer answer: cardinal systems (the generalized form that approval is part of) is able to capture more information about voter preference. It is easiest to see this with something like range voting. You may be familiar with a question like "Do you agree or disagree with x statement?: strongly disagree, disagree, neutral, agree, strongly agree." That is range. Approval would be "Do you agree or disagree with x statement?: Disagree, agree". Conversely FPTP would present you many statements and you have to pick the one you agree with the most, even though you may agree with many in the list. You can probably see here that you are actually able to capture a fair amount of preference information through this, thus allowing to elect representatives that more accurately represent the population as a whole.