Get the current state of the mouse by using GetState.

MouseState mouseState = Mouse.GetState();

Get the current screen coordinates of the mouse from X and Y.

int mouseX = mouseState.X;
int mouseY = mouseState.Y;

Using Viewport.Unproject, determine points in world space on the near and far clip plane. For the point on the near plane, pass a source vector with x and y set to the mouse position, and z set to 0.

For the point on the far plane, pass a source vector with x and y set to the mouse position, and z set to 1.

For both points, pass Unproject the current projection matrix, the view matrix, and a translation matrix for the point (0,0,0).

Vector3 nearsource = new Vector3((float)mouseX, (float)mouseY, 0f);
Vector3 farsource = new Vector3((float)mouseX, (float)mouseY, 1f);

Matrix world = Matrix.CreateTranslation(0, 0, 0);

Vector3 nearPoint = GraphicsDevice.Viewport.Unproject(nearsource, 
    proj, view, world);

Vector3 farPoint = GraphicsDevice.Viewport.Unproject(farsource, 
    proj, view, world);

Create a Ray starting at the near point and pointing toward the far point.

// Create a ray from the near clip plane to the far clip plane.
Vector3 direction = farPoint - nearPoint;
direction.Normalize();
Ray pickRay = new Ray(nearPoint, direction);

Use Intersects to loop through the objects in the world to check whether the Ray intersects each object.

If the Ray intersects an object, check whether it is the closest object intersected so far. If it is, store the object and the distance at which it was intersected, replacing any previously stored object.

When you completely loop through the objects, the last object stored will be the closest object underneath the area the user clicked.