#The Law
Bestiality is not a crime, per se, in five states, although in some cases it may constitute animal cruelty.
In Colorado (where the enactment was fairly recent), bestiality is per se an act that constitutes cruelty to animals, but some states classify it is a form of immoral and indecent conduct, focusing on the immorality of the perpetrator rather than on the alleged harm to the animal.
These laws are generally enforced by local prosecuting attorney's offices (a few states, such as Florida, assign the function to a state agency instead), which generally do not have coherent policies on the topic and play mostly a behind the scenes role in the legislative process, usually through a statewide lobbying organization for prosecuting attorneys generally.
#Why Are These Law Enacted?
Basically, bestiality laws are reactions to moral panic:
a feeling of fear spread among a large number of people that some evil threatens the well-being of society. A Dictionary of Sociology defines a moral panic as "the process of arousing social concern over an issue – usually the work of moral entrepreneurs and the mass media".
The legislative history of the Ohio example, detailed by @Fizz illustrates the moral panic narrative of these enactments, following a repeal of many such laws in connection with the repeal of anti-sodomy laws.
In some cases, anti-sodomy laws that included bestiality where repealed in connection with state overhauls of their criminal statutes modeled on the Model Penal Code (1962) which served as a template for state legislation (at least in part) in two-thirds of U.S. states, which did not include a bestiality offense. Many of the legal revisions modeled on the Model Penal Code took place in the early 1970s. But, some of those states later enacted bestiality laws as noted in the answer of @Fizz regarding the bimodal distribution of enactment dates.
Other anti-sodomy laws that happened to also include bestiality were repealed as a housekeeping matter because the case Lawrence v. Texas, 539 U.S. 538 (2003) that held a law banning consensual sodomy between adults to be unconstitutional. Conservative Justice Scalia's dissent in that case argued that if the court was not prepared to validate laws based on moral choices as it had done in Bowers v. Hardwick, 478 U.S. 186 (1986) (which Lawrence overruled), state laws against bigamy, same-sex marriage, adult incest, prostitution, masturbation, adultery, fornication, bestiality, and obscenity would not prove sustainable.
An account from 2005 in Colorado:
Speaker of the House Andrew Romanoff sums up this point succinctly in talking about another issue that occurred last spring when Republican Reps. Keith King and Jim Welker started comparing homosexuality to increased crime rates and later bestiality. Said Romanoff: “We’re talking about the budget, and they’re talking about bestiality.”
The bestiality prohibition in Colorado was enacted two years later, but as a form of cruelty to animals rather than as a form of immorality law, a compromise reached to allow pro-gay rights legislators to avoid attacks claiming that they were pro-bestiality.
The motives of the proponents of the law sums up some of the motive of recent legislative action on the issue, which is that a focus on criminalizing bestiality flows, in part, from an effort to obtain legislative buy-in to an agenda that also includes opposition to gay rights.
Much of the conservative opposition to gay rights in turn derives from declining marriage rates and increasing divorce rates, especially for working class adults, which conservatives have framed as a decline in and threat to a societal commitment to "family values" such as marriage between one man and one woman, and sex as confined to marriage. (The actually cause has more to do with the stagnant or declining economic prospects of working class men relative to the rising economic prospects of working class women.)
Purportedly this traditional marriage concept has roots in the Bible, even though this ideal is does not have Biblical support and is instead a considerably later development (e.g. the Biblical Hebrews were polygamous). Bestiality laws, of course, do have a Hebrew Bible (a.k.a. Old Testament) precedent, e.g. in Exodus 22:19 which states:
Whoever lies with an animal shall surely be put to death.
#Scholarship
There is a fair amount of legal or partially legal/political scholarship on the topic as well as the evolving behavioral science literature. Some of the most cited and notable include the following (in reverse chronological order, you will need to find your own links in some cases):
Christopher Hensley, Suzanne E. Tallichet, Stephen D. Singer, "Exploring the Possible Link Between Childhood and Adolescent Bestiality and Interpersonal Violence" 21(7) Journal of Interpersonal Violence (2006) doi.org/10.1177/0886260506288937
A.M. Beetz, A.L. Podberscek (editors), "Bestiality and zoophilia: sexual relations with animals" (2005) (Book length anthology).
RE Jenkins, AR Thomas, N Loudonville-Abgefragt "Deviance online: Portrayals of bestiality on the Internet" Center For Social Science Research Conference Presentation (2004) (open access full text).
Jens Rydström, "Sinners and Citizens: Bestiality and Homosexuality in Sweden, 1880-1950" (2003) (Book).
Peter Singer, "Heavy petting." Nerve (2001).
Peter Singer, "Clarification [of] the circumstances and intent of [my] review of Midas Dekkers' book Dearest Pet." Princeton University press release, April 14, 2001.
P. Beirne, "Peter Singer's``Heavy Petting'' and the Politics of Animal Sexual Assault" Critical Criminology (2001).
P. Beirne "On the sexual assault of animals: A sociological view." in A. Creager and B. Jordan (eds.), "Identity and Alterity: Essays on the Human/Animal Boundary" (2001).
M.H. Pierson, "Dark Horses and Black Beauties." (2000).
Jürgen Habermas, "Bestiality and humanity: a war on the border between legality and morality" 6(3) Constellations 263 (1999).
John M. Murrin, "'Things Fearful to Name': Bestiality in Colonial America" Pennsylvania History: A Journal of Mid-Atlantic Studies 8 (1998) (open access full text).
P. Beirne, "Rethinking bestiality: Towards a concept of interspecies sexual assault" Theoretical Criminology (1997).
C.J. Adams, "Woman-battering and harm to animals." in C.J. Adams and J. Donovan (eds.), "Animals and Women: Feminist Theoretical Explorations" 55 (1995).
M. Dekkers, "Dearest Pet." (translated by P. Vincent) (1994).
J. Salisbury, "The Beast Within: Animals in the Middle Ages." (1994).
B. Noske, "Hoe heet is een ezelin?" 21 Opzij, Feministisch Maandblad 26 (1993).
J. Liliequist, "Peasants against nature: Crossing the boundaries between man and animal in seventeenth-and eighteenth-century Sweden." 1(3) Journal of the History of Sexuality 393–423 (1991)
G. Rubin, "Thinking sex: Notes for a radical theory of the politics of sexuality" Social perspectives in Lesbian and Gay Studies (1984) (in Spanish).
T. Regan "The Case for Animal Rights." (1983).
Dolf Zillmann, Jennings Bryant, Rodney A. Carveth, "The Effect of Erotica Featuring Sadomasochism and Bestiality on Motivated Intermale Aggression" 7(1) Personality and Social Psychology Journal (1981). doi.org/10.1177/014616728171023
E.W. Monter, E.W. "Sodomy and heresy in early modern Switzerland." 6 (1/2) Journal of Homosexuality 41 (1980).
Peter Singer, "Animal Liberation." (1975).
Hensley and Zillman extend the theory that bestiality like other forms of animal cruelty is predictive of psychopathy and violence against humans.
Peter Singer is the most prominent advocate for only making it a crime when there is actual cruelty to the animal on a case by case basis, with criticism of him in papers commenting on his work. Critics on the animal cruelty front are largely arguing that animals can't meaningfully consent.
The links, Monter and Rydström, also indicate the linkage of LGBTQ issues to Bestiality, in general, in legal/political motive.
Several look at historical status like Murrin, Monter, Rydström and Liliequist are applicable to the reasons for older wave statutes, basically moral ones motivated by "disgust" and traditional morality.