Does neoclassical economics explain or assume private property rights? I understand that neoclassical economics is empirical, in contrast to the a priori base of Austrian economics. I therefore wonder if neoclassical economics also explains, rather than assumes, private property rights?
Intuitively private property rights offer a greater incentive to invest than mere use rights, such as leases.
Private property also reduces transaction costs by internalising, rather than contracting, activities (theory of the firm).
Is this how private property is explained in economics?
It is true that mainstream economics (which includes neoclasicall economics) is empirical but note that any explanation can  only arise from theory (whether theory is explicitly formulated or only implicit in the background), and any theory is based upon assumptions. Empiricism does not mean you don't build theoretical models with assumptions, it means you go and test those theoretical models rather than just rely on deductive reasoning alone.
For example, theory of relativity is explanation of gravitational attraction deduced from some assumed premises. This theory can be tested by observing whether light bends around massive object. However, note it does not work the other way around. Observing that light bends around massive objects does not provide any explanation for gravitational attraction - on its own it is just a empirical curiosity. This is not to say that one might not first observe something and then develop theory, but point is that explanations themselves can't be empirical.
That out of the way these are common explanations for property rights in mainstream economics with references that also contain empirical examples:

Private Property Solves the Tragedy of Commons: The tragedy of the commons is a situation that arises when goods/resources are commonly owned where individuals, acting independently behave contrary to the 'common good' by depleting the shared resource through their collective action. This happens because if everyone owns let's say fish in the lake, then everyone has incentive to extract those fish as soon as possible otherwise someone else will do so. This is an example of prisoner dilemma type situation. Hence, excessive pollution, overfishing, deforestation and other such examples are observed virtually  every time there is a lack of well defined property rights. Property rights (where possible to apply) solve this issue because if one person has right over some resource they have an incentive to exploit it sustainably  (see Mankiw Principles of Economics pp 218-223).  This is why ivory trade makes elephants almost extinct (as they are 'commons' in many places), while caws are not endangered even if they are much more intensely used for food or skin or why rainforests are disappearing while privately owned forests are managed sustainably.
Private Property Encourages Investment: This was already correctly mentioned in your answer. Some altruistic preferences/utility aside people will invest less in something when they will not reap all the benefits. This is why it is very uncommon to see people making large investment in leased car or property. If you rent a house why would you renovate it just so later on when your lease expires someone else is going to enjoy the investment.
This is even more important in context of investing in factors of
production like capital goods. This is crucial for economic development. Economic growth arises from investment (see Barro &amp; Sala-i-Martin: Economic Growth 2nd ed). Empirically the above two are very important as there are virtually no societies that would achieve sustained economic development without property rights and other inclusive institutions, while a common factor among societies that remain  poor lack secure property rights (see Acemoglu &amp; Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights Encourage Productivity: Private property rights provide incentive for production and encourage productivity. For example, if you are a farmer but there are no property rights so anyone can just walk on your farm and take any produce they want this destroys any incentive for people to produce anything beyond immediate consumption (again see Acemoglu &amp; Robinson: Why Nations Fail? and sources cited therein).
Private Property Rights are Prerequisites for Well Functioning Markets: This is connected to the two point above, but I list it as a separate reason as it goes little bit further. Lack of well defined and secured property rights is actually considered market failure and in fact government providing well defined and secure property right is considered to be necessary intervention to make markets work. As argued by Stiglitz (Economics of the Public Sector pp28):


An important activity of the government, but one that accounts for very little expenditure, is the establishment of the legal framework within which firms and individuals can engage in economic interactions. ... Without laws defining property rights, only the exercise of force would stop one individual from stealing from another. Washout the ability to protect property, individuals would have little incentive to accumulate assets. Needless to say, economic activities would be severely restricted.

Later on pp 77:

But even for markets to work, there needs to be a government to define property rights and enforce contract.

In fact Mueller in Public Choice III pp 10 argues enforcement of property rights is one of the reasons for people developing 'constitutional contracts' and moving beyond state of Hobbesian anarchy toward having some sort of organized state.

Private Property Reduces Transaction Costs: This was again already mentioned in your question and it is also one of the reasons you see in the literature. Often property rights can be used to provide better incentives for people to behave in certain way. For example, a firm my require employee to own certain asset (e.g. car) rather than provide it to ensure that its taken good care, which might be cheaper than having some monitoring in place  (see  Williamson The Economics of Transaction Costs and sources cited therein).

There might be other explanations,  that I have missed and hopefully someone other users can fill those gaps, but I think I listed all the major ones.