Lets say you split the image by 13x13 (SxS) cells. For each grid cell, you predict 5 (older YOLO) or 9 (newer versions) bounding boxes.
These (anchors) are supposed to have various shapes and scales, so they vertical, horizontal and squared rectangle at small scale, and middle scale and bigger scale.
So for each cell (13x13) and for each anchor (9) , you predict (width offset ,height offset,center x,center y, objectness, class1 ,class2,... class k ). You actually predict scaling of the width and height of the box, so you can end up with variable sizes.
Ojectness marks if there is any object in this cell for this anchor, if not, it is discarded later.
Since you end up with lot of predicted bounding boxes of various shapes, you need to filter them out, this is called non maximum supression.
For each anchor you take class with highest confidence.
So to answer your questions, If an object is in multiple cells, it is contained in multiple predicted boxes, and you take the one with maximum confidence and discard others.
Each cell does not see only part of the image, it sees the whole image. Before you get to 13x13 cells, you have 50 or more layers of conv & max pooling layers, which gives each cell receptive field of the whole image.