TL;DR

Because that's their goal, and it works.

Silver Spring Monkeys

First up, the Silver Spring case is an interesting starting point for PETA, because Edward Taub's work with the monkeys turned out to be groundbreaking research into neuroplasticity, and led to the development of constraint-induced movement therapy, which is used to help stroke victims recover. Additionally, PETA's undercover investigator, Alex Pacheco, waited until Taub was on vacation to "document the lab" (jump down to Silver Spring Monkeys), giving Taub a stronger defense that Pacheco intentionally trashed the lab before documenting it. This defense was more plausible because the state prosecutor joined PETA, the monkeys were housed with PETA's president by the state, and the monkeys mysteriously disappeared from her house and returned with no explanation.

And true to form, PETA didn't care about the research, didn't care about the goals, didn't care about anything other than the welfare of the monkeys under experiment. Whether you agree with PETA (that intentionally removing sensation from limbs was unethical) or Taub (that the research was important work for people who have nerve damage), the Silver Spring case was a complete harbinger of PETA's all-or-nothing tactics and unwillingness to accept nuance.

ALF

Not long after Silver Spring, the first American Animal Liberation Front cell was formed. PETA, ostensabily, is non-violent. ALF claimed to start out non-violent, but is considered a terror group by the FBI, and had ties both to the Earth Liberation Front (ELF) and PETA. PETA gave money to ALF, acted as the spokesgroup for the front, and there has been documented cross-pollination between the two groups. The ties between PETA, ALF, and ELF are covered in this Senate committee hearing report.

Note: PETA's president, Ingrid Newkirk also wrote in ALF's "newsletter". The ties were never hidden, nor meant to be hidden.

Ill-considered publicity stunts

Let's first start that their reputation for Ill-considered publicity stunts is well earned, and comes from the fact that it's the point of the group. From interviews given to the New Yorker in 2003, PETA's president Newkirk invited the reporter into their weekly war council:

Each week, Newkirk holds a kind of war council: she gathers two dozen of her top strategists around a square table in the second-floor conference room to plot their next moves, and while I was in Norfolk she invited me to join them. Jason Baker, who runs the peta operation in Hong Kong (there are also offices in England, Germany, Holland, and India), presented a slide of a new advertisement he was preparing for the Asian market to publicize the plight of elephants. It is a picture of a naked woman, shackled and in chains. (The woman, Imogen Bailey, was recently voted Australia's sexiest model.) "We are going to put whip marks on her back,'' Baker explained to approving mutters, "and, if it works visually, tears in her eyes." Newkirk stared at the picture for a minute and then shook her head. "She looks like she's pouting,'' she said. "It's too sexy. We need to make her look terrified." Baker promised to take care of it.

Remember - this is Newkirk's firsthand account of her own thought processes. The "ill conceived publicity stunts" happen because PETA morphed into a publicity stunt generator. Between Silver Springs and our 20 year cutoff in 2001, PETA's growing budget wasn't spent on investigation, it started being spent more on PR than investigation (as seen in the oldest IRS Form 990 I could find, from 2004).

Maybe someone in later comments can add more specifics pre-2001 to some of the publicity stunts, but a notable 2001 stunt was lamenting orphaned animals in the wake of 9/11 and demanding that Rudy Giuliani put together an animal-focused task force.

Impact

PETA's impact in animal rights cuts many ways:

Animal Cruelty Enforcement: their activism has raised the profile of animal rights and inspired many other mainstream animal-focused groups that are more normal. They've inspired new animal cruelty laws, increased enforcement of those laws. For all the focus on "PETA runs stupid campaigns that anger everyone", Animal Cruelty is simply an issue that was seen differently and more importantly in 2000 than in 1980, and is seen as more important now than in 2000.

Anti-Animal Investigation Enforcement: Their tactics (such as infiltrating laboratories and big ag sites) also have led to acts such as the Animal Enterprise Protection Act in 1984 to allow the DoJ to target animal rights groups for infiltrating and damaging agricultural, and efforts by research and agricultural issues (into the last 20 years) to criminalize other activities. Ag gag laws actually help PETA, because to people who care about animals, it makes Big Ag look worse just by pushing for them.

Consultation and Improvement: As PETA is a PR engine, companies quickly realized that they needed to combat PR with PR, but also to actually improve. For example, the agriculture industry began hiring consultants such as Temple Grandin to make their operations less abusive. Research into agricultural and animal bioethics increased through the period (leading to dedicated research groups at major universities now, such as the Candace Croney Research Group at Purdue University). In 2000, McDonalds became the first fast food company to enforce supplier standards around treatment of hens (a practice that has since accelerated and become more industry standard).