The Product Owner should care that the Scrum Team adheres to its Definition of Done because:
It helps to forecast the team's productivity over time. By having a clear and consistent Definition of Done, the Product Owner can measure the amount of work that the team can deliver in each Sprint and plan accordingly.
It affects the product's total cost of ownership. By ensuring that the work done by the team meets the quality standards and expectations of the stakeholders, the Product Owner can reduce the risk of technical debt, rework, defects, and maintenance costs in the future.
Reference:
Scrum Guide 2020, page 10: "The Definition of Done is a formal description of the state of the Increment when it meets the quality measures required for the product." Scrum Guide 2020, page 11: "The Product Owner is accountable for effective Product Backlog management, which includes ... ordering Product Backlog items; and, ensuring that the Product Backlog is transparent, visible and understood." Scrum Guide 2020, page 14: "The Definition of Done creates transparency by providing everyone a shared understanding of what work was completed as part of the Increment. If a Product Backlog item does not meet the Definition of Done, it cannot be released or even presented at the Sprint Review. Instead, it returns to the Product Backlog for future consideration."

Explanation
* The Scrum Team is a self-organizing and cross-functional team that delivers valuable products in an agile way. The Scrum Team consists of one Product Owner, one Scrum Master, and Developers. They are all accountable for creating a valuable, useful, and potentially releasable product Increment each Sprint.
* The Scrum Team is responsible for making sure everyone on the Scrum Team does their tasks for the Sprint. The Scrum Team does not rely on any external authority, manager, or leader to assign or monitor their work. The Scrum Team collaborates and coordinates their work as one unit, without any hand-offs or silos. The Scrum Team also inspects and adapts their work based on empirical evidence and feedback.
* The Product Owner is accountable for maximizing the value of the product resulting from the work of the Scrum Team. The Product Owner is responsible for managing and refining the Product Backlog, collaborating with the stakeholders and the Developers, and ordering the items in a way that best achieves goals and missions. The Product Owner represents the interests of everyone with a stake in the product and ensures that the Scrum Team works on the right things at the right time.
* The Developers are accountable for creating a "Done" Increment that meets the Definition of Done each Sprint. The Developers are responsible for planning and executing the Sprint Backlog, designing and building the product functionality, testing and improving the product quality, and delivering a potentially releasable Increment. The Developers work closely with the Product Owner to understand and clarify the Product Backlog items, provide feedback and estimates, and suggest improvements and innovations.
* The Scrum Master is accountable for establishing Scrum as defined in the Scrum Guide. The Scrum Master is responsible for promoting and supporting Scrum as defined in the Scrum Guide. The Scrum Master does this by helping everyone understand Scrum theory and practice, both within the Scrum Team and the organization.
* The other options are not valid or relevant answers for who should make sure everyone on the Scrum
* Team does their tasks for the Sprint. They are either nonexistent or irrelevant roles in Scrum. They are:
* The Project Manager: There is no "Project Manager" role in Scrum. Scrum is a framework for developing, delivering, and sustaining complex products, not projects. A project is a temporary endeavor with a fixed scope, time, and cost. A product is a valuable solution that evolves over time to meet customer needs and market conditions. A project manager is a role in traditional project management, a discipline for planning, executing, and controlling projects. A project manager coordinates and controls the activities of the project team and the stakeholders and ensures that the project meets the quality standards and expectations.
* The Product Owner: The Product Owner is not responsible for making sure everyone on the Scrum Team does their tasks for the Sprint. The Product Owner is responsible for maximizing the value of the product resulting from the work of the Scrum Team. The Product Owner does not assign or monitor tasks to the Developers. The Product Owner collaborates with them to clarify and refine the Product Backlog items and their acceptance criteria.
* All of the above: This is not a valid answer because it includes two invalid options: The Project Manager and The Product Owner.
References:
* Scrum Guide: https://www.scrumguides.org/scrum-guide.html
* Scrum Team: https://www.scrum.org/resources/what-is-a-scrum-team
* Product Owner: https://www.scrum.org/resources/what-is-a-product-owner
* Developers: https://www.scrum.org/resources/what-is-a-developer-in-scrum
* Scrum Master: https://www.scrum.org/resources/what-is-a-scrum-master

It is true that the Sprint Backlog is a result of Sprint Planning, and it includes the Sprint Goal. This is because:
* Sprint Planning is an event where the Scrum Team plans for the upcoming Sprint. The purpose of Sprint Planning is to align the entire Scrum Team around a common goal and a plan for delivering an Increment that meets that goal.
* The Sprint Backlog is the set of Product Backlog items selected for the Sprint, plus a plan for delivering them as a "Done" Increment. The Sprint Backlog is owned by the Developers who use it to organize and manage their work during the Sprint.
* The Sprint Goal is a short-term objective that provides guidance and focus to the Scrum Team throughout the Sprint. It is a flexible and negotiable commitment that can be adjusted as more is learned throughout the Sprint.
* The Sprint Backlog is created by the collaborative work of the entire Scrum Team during Sprint Planning. The Product Owner proposes how the product could increase its value and utility in the current Sprint. The Developers discuss how they can deliver a "Done" Increment that meets this proposal. Together, they define a Sprint Goal that summarizes why the Sprint is valuable to stakeholders. Then, they select enough Product Backlog items from the top of the ordered Product Backlog to satisfy the Sprint Goal. Finally, they create a plan for how they will deliver those items as a
"Done" Increment.
References:
* [Scrum Guide], page 14, section "Sprint Planning"
* [Scrum Guide], page 15, section "Sprint Backlog"
* [Scrum Guide], page 15, section "Sprint Goal"