At the time of writing, I'm not aware of any books that are very specifically about topological data analysis (TDA), apart from the collected papers in [Topological Methods in Data Analysis and Visualization](http://www.amazon.com/gp/product/3642150136/ref=s9_simh_gw_p14_d1_i5?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-2&pf_rd_r=1WT77Y4V08T1Y8K44GTF&pf_rd_t=101&pf_rd_p=1688200382&pf_rd_i=507846) and its two sequels, but there are a handful on computational topology that contain valuable background and details for TDA. Gurjeet has already mentioned Afra Zomorodian's [Topology for Computing](http://www.amazon.com/Computing-Cambridge-Monographs-Computational-Mathematics/dp/0521136091/ref=pd_sim_sbs_b_3?ie=UTF8&refRID=1F8YTMKRRC1S14Q6VBG8). Others include:

 - [Computational Topology](http://www.amazon.com/Computational-Topology-Herbert-Edelsbrunner-Harer/dp/0821849255/ref=pd_sim_sbs_b_1?ie=UTF8&refRID=0QCZ3AZTG3KEKJJNQ6A1) by Edelsbrunner & Harer
 - [A Short Course in Computational Geometry and Topology](http://www.amazon.com/gp/product/3319059564/ref=s9_simh_gw_p14_d0_i5?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-2&pf_rd_r=1SZG9QZ9P8D94VH13K2Q&pf_rd_t=101&pf_rd_p=1688200382&pf_rd_i=507846) by Edelsbrunner
 - [Computational
 Homology](http://www.amazon.com/Computational-Homology-Applied-Mathematical-Sciences/dp/1441923543/ref=pd_sim_sbs_b_4?ie=UTF8&refRID=1AE8B034Z40PPZ8YNYC0)
 by Kaczynski et al

At the moment, knowledge of statistics does not appear to be a prerequisite, although there is some interesting work in that direction at CMU: http://www.stat.cmu.edu/topstat/. It is helpful to be comfortable with multivariable calculus, linear algebra, introductory abstract algebra (especially group theory) and basic point-set topology. Prior acquaintance with algebraic topology and manifolds would be even better. For comparison purposes, it may be interesting to look into clustering algorithms such as [$k$-means](http://en.wikipedia.org/wiki/K-means_algorithm) and [hierarchical clustering](http://en.wikipedia.org/wiki/Hierarchical_clustering).

You may want to take a look at Peter Saveliev's draft text [Topology Illustrated](http://inperc.com/wiki/index.php?title=Topology_Illustrated) (which is indeed liberally and helpfully illustrated, so the title is accurate) with its emphasis on homology, and Robert Ghrist's [Elementary Applied Topology](http://www.math.upenn.edu/~ghrist/notes.html) for a broad-ranging invitation to applied topology. Michael Robinson's [Topological Signal Processing](http://www.amazon.com/Topological-Signal-Processing-Mathematical-Engineering/dp/364236103X/ref=sr_1_1?s=books&ie=UTF8&qid=1413524712&sr=1-1&keywords=topological+signal+processing) could also be of interest.