introduction to Chip Seq
The basic steps of ChIP-Seq are:


Crosslinking proteins to DNA - this fixes the proteins in their natural positions
Nuclease digestion - this removes regions that are unbound to protein; nucleases are sterically hindered from digesting protein-bound DNA
Immunoprecipitation - this allows isolation of the target protein by binding it to a selective antibody
Sequencing to determine the bound DNA sequence
Mapping the reads to the reference genome


Why so many reads? They are taking many reads from different cells. Why are the reads so short? Many modern next-gen sequencing methods are limited to ~30 bp reads. Thus, the reads you see are not actually the entire protected sequence but rather just a shorter fragment from the 5' end (where sequencing starts from). The why behind this is explained by Jiang and Pugh, 2009 (Nucleosome positioning and gene regulation: advances through genomics):


  Three DNA sequencing technologies have been used to map nucleosomes. Pyrosequencing using the Roche 454 GS20/FLX
  sequences nucleosomel DNA end-to-end, allowing both nucleosome
  borders to be linked in a single sequence. This method provides the
  greatest mapping accuracy, particularly in genomic regions of low
  complexity. By contrast, other platforms, such as those provided by
  the Illumina-Solexa Genome Analyzer and Applied Biosystems SOLiD,
  generate only 25-35 by sequence tags, requiring both nucleosome
  borders to be inferred. Nonetheless, these short-read technologies
  produce >100 times the number of sequence tags at a similar cost as
  the long-read technologies, and so the short-read technology is
  currently the only practical technology for mapping nucleosomes in
  large genomes. The higher tag count of the short-read technology
  enhances mapping accuracy and thus provides a practical way of
  mapping nucleosomes.


Why are the reads overlapped? I can think of a few explanation for that. For starters, Protein-DNA binding is variable in its specificity. In other words, a given protein might bind at the exact same site every time or it may bind 'around' a site or it may bind randomly. A protein that binds an exact sequence would be expected to produce reads that are more in phase that one that binds more variably. Next, the nuclease digestion process is random (or at least expected to be). Thus the DNA cleavage point relative to the protein is variable and your reads start from different points. Again, from Jiang and Pugh:


  At any given genomic locus, the preferential positioning of nucleosomes--called phasing--can be described. At most loci, there is an approximately Gaussian (normal) distribution of nucleosome positions around particular genomic coordinates, ranging from ~30 by for highly phased nucleosomes to a random continuous distribution throughout an array. How much of this variation is due to genuine positional heterogeneity and how much is an artefact that is caused by overtrimming or undertrimming of the DNA at nucleosome borders by micrococcal nuclease during sample preparation remains to be determined. 


Does that answer your question? Basically the reads are overlapped because there are many of them and they may differ either because of actual variance in the DNA binding site (which is biologically significant) or because of variance in the DNA digestion site (which is a methodological flaw). As with any biological experiment, you must perform many replicates to statistically determine the source of the variance and this is done by taking many reads. I recommend you read the article I quoted from, it is a good overview of histone mapping with ChIP and is likely also applicable to what you're studying.

Also, do you know why there are two sets of reads (red and blue)? I would venture a guess that they are taken from opposite strands. In other words, they are reads of the same DNA region taken from different directions. That might help clarify what's going on.

To conclude:


DNA is not extracted from a single cell. In usual procedures it is
always from a collection of cells or from tissues.
The overlap in TF associated reads arises because of the TF bound
sequence coming from different cells in this collection.
Each DNA piece is amplified and clusters are formed in the NGS
sequencers (for the ease of detection);
In a homogeneous cell collection i.e. all cells are of same type, the
variability in the read position usually arises because of
flexibility of binding sites (i.e TF wont exactly bind to at a
certain locus). Also, as previously mentioned the variability can
arise during the technical procedures.
In case of heterogeneous tissues the variability arises because of
different biochemical status of different cells. Strictly speaking
there is always some heterogeneity even in an apparently homogeneous
tissue but tissues such as brain are highly heterogeneous
In the case of nucleosomes the positioning is not exact because the
sequence contribution to the position is very minimal. Certain
nucleosome positions are tightly maintained for e.g. around the TSS
(transcription start sites) whereas in other regions the positioning
is fuzzy because of the dynamic nature of the chromatin.