Why must DNA be packed into chromosomes during mitotic phase?
A chromosome is simply a length or segment of DNA. Bacteria have few structural proteins on their DNA, and they have one circular chromosome. In humans, before DNA replication, the nucleus contains 46 strands of DNA, i.e. chromosomes (22 chromosomes in two copies and usually two X or one X and one Y for males and females, respectively). All chromosomes are DNA that is bound by histone proteins that are organised in nucleosomes, approximately every 147 base pairs.

What you are referring to is a condensed chromosome in metaphase, which confusingly in humans is actually an X shape consisting of 2 identical chromatids, structurally condensed and joined at the centromere to form a duplicated chromosome.

The duplicated chromosomes must condense and join before the cell can divide because a tangle of 96 chromosomes as chromatin would be almost impossible to organise and separate properly. The joining of identical chromosomes and compacting process allows all the 46 compacted and duplicated chromosomes (i.e. 96 chromatids) to simply align along the spindle apparatus that forms in centre of the cell during mitosis, and have separation of the chromatids during anaphase so that each of the new cells has a full set of 46 chromosomes.