It's not so much a government agency as it is a public corporation. In the days when the postal service was founded, large corporations were rare enough that they had to be created and initially funded by the government. Now we have financial markets and concepts that allow for the streamlined creation of large corporations, but it's closer to the modern concept of a utility than anything else.

> Why does the postal service need to turn a profit?

It doesn't have to turn a 'profit' in the literal sense, but it needs to make more money than their current spending because they have to pay future pensions.

How else will it grow and expand to better serve their customers?

Yes? I think lots of government agencies that deliver services to paying customers are expected to cover their expenses. It's true that some government activities provide, say, non-excludable public goods using tax money, and so are not expected to collect sufficient revenue through distribution, but other government activities are intended to provide a service that can be valued and efficiently purchased by consumers.

For instance, my family's street recently got a public sewer system (to replace septic systems), and the city charged homeowners one-time and recurring fees to cover the expenses of both installation and operation. If the city could not get the homeowners to pay enough to cover the cost of the sewer system, which was a real risk, the system would have been considered a mistake (and a project with similar economic prospects would likely been avoided in the future).