Some details of the class have been removed. In particular, the constructor dynamically allocates the DataStructure object and the destructor deallocates it. If, during a move, you just copied the pointer from one Widget to another, both Widgets would have pointers to the same allocated DataStructure object. Then, when those objects are destroyed, they would both attempt to delete it. This would give undefined behaviour. To avoid this, the Widget that is being moved from has its internal pointer to set to nullptr.
This a standard pattern when implementing a move constructor. You want to move ownership of some dynamically allocated objects from one object to another, so you need to make sure the original object no longer owns those allocated objects.
Diagrammatically, you start off with this situation, wanting to move ownership of the DataStructure from one Widget to the other:
┌────────┐ ┌────────┐
│ Widget │ │ Widget │
└───╂────┘ └────────┘
┃
▼
┌───────────────┐
│ DataStructure │
└───────────────┘
If you just copied the pointer, you'd have:
┌────────┐ ┌────────┐
│ Widget │ │ Widget │
└───╂────┘ └───╂────┘
┗━━━━━━━━┳━━━━━━━┛
▼
┌───────────────┐
│ DataStructure │
└───────────────┘
If you then set the original Widget pointer to nullptr, you have:
┌────────┐ ┌────────┐
│ Widget │ │ Widget │
└────────┘ └───╂────┘
┃
▼
┌───────────────┐
│ DataStructure │
└───────────────┘
Ownership has successfully been transferred, and when both Widgets can be destroyed without causing undefined behaviour.