Frequently cities and counties install improvements like curbing or sewers in a neighborhood. Rather than taking money out of general revenues to pay for these improvements, the cities and counties assess the owners of property in the neighborhood for the improvements. Generally these assessments are paid in installments over several years over several years as part the yearly real estate tax bill. When a house is sold, there is a question as to whether the buyer or the seller should pay the cost of the assessments. The provision you quoted is a provision in a purchase and sale contract designed to deal with this question.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.