tl;dr

You can't (or at least if people are actually following Google's internal policies to not disclose specific rejection reasons, you shouldn't be able to).

"I don't know if it was the total scoring on the interviews or something on my resume that made the HC believe that I wasn't a good fit for the role"

Your resume got you the interview. Once interviewed, with Google your resume is largely irrelevant and hiring decisions are largely based on interview performance and internal references (job hopping is the one thing on your resume that HCs may consider that would result in your rejection at this point).

"Any ideas on how I could get more details?"

Yes. Call your Recruiter back, but don't ask "why was I rejected" or "can I get more details". They won't/can't answer that. Instead say, "I'd like to do some studying before reapplying again, and I'm trying to determine where I should best devote my energies. Do you have any thoughts on what I should focus on most? Coding? Algorithms? Software design?" (or if you applied for something other than SWE, substitute the core proficiencies you were interviewed on for your specific function)

By isolating this question from the HC/rejection decision, you *might* be able to get an answer from your Recruiter. If not (and you don't feel that it's just because the Recruiter is too lazy to look up their notes) invariably you may have been rejected because you just didn't do quite well enough across the board ("insufficient enthusiasm from interviewers" is the most common rejection reason for borderline cases).

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[Note: The rest of this answer addresses some ancillary points in the original question, as well as some misconceptions -- relative to Google specifically -- contained in other respondents' answers]

It's unclear what "engineering role" you were being considered for. The vast bulk of Google's engineering hiring is for SWE roles, and while for them candidates are targeted for a particular product area/team -- hiring authority is vested in Hiring Committees (unlike at many? most companies, "Hiring Managers" at Google don't have hiring authority).

Hiring Committees are tasked with assessing fit not just for a specific team/project, but instead for Google Engineering overall (Google encourages internal mobility between projects and Hiring Committees exist there to ensure that an engineer will be a fit beyond for just a narrow team/area). There are some rare exceptions to this -- but typically only for manager/director roles or for unique areas such as in Google X or hardware design.

To be a bit more blunt that your Recruiter may have been willing to be -- based on the scenario you described, in all likelihood you were actually rejected (i.e. "no hire"decision) when the HC reviewed your packet. However, your case was borderline enough (or your internal referral was strong enough) that your Recruiter -- hoping to turn things around and get you hired (Recruiters' performance is largely measured by the number of hires they make) -- found a Hiring Manager who was more positive than the Hiring Committee itself, and invited that Manager to come to the HC in hopes that s/he would convince the HC to reverse their earlier decision.

"Appeals" like this are sometimes successful, but rarely (if I had to make up a number, I'd guess 10-15% of such appeal cases turn around into a hire).

"The position was still open after I received the rejection."

That doesn't mean much of anything. The vast majority of engineering job reqs at Google (especially in MTV) are multi-hire reqs, and rarely are "closed" as a result of a hire (ex: 1000's of hires are made each year under the "Software Engineer - Mountain View" req, which has been open continuously since Google's inception). Accordingly, candidates aren't compared to other candidates and Google instead weighs each candidate individually against their "bar".

Lastly, "I thought the interviews went fairly well (one coding error but nothing that I couldn't answer)

In many cases, candidates simply don't recognize that their answers were actually poor, and that they didn't do as well as they had thought. However, in a lot of cases, a candidates' answers may actually all have been good, but they simply took longer to get to the solution than the interviewer(s) felt was necessary (ex: "Candidate came up with a great solution, but it took the full 45 minutes of our interview to do something most good candidates take only 5 minutes to come up with"). Not sure if either of these apply to you specifically, but just wanted to throw it out there...

"I asked him how long would I have to wait before applying for a similar role and he said that I did't have to wait to apply for a new position."

I don't know what role you applied to, but I'm not sure why your Recruiter would have told you that. Google's policy is to have a minimum of 12 months elapse before reconsidering a candidate for the same or largely similar position again. If someone interviewed for SWE and wanted to apply for Product Manager -- there's no wait time required. However, if someone applied for "SWE - Mountain View" and wanted to apply for "Front End SWE - New York" (largely the same requirements), the 12 month rule applies

(18 months is actually preferred, and even longer the deeper someone is into their career. If someone, for example, has been working as a SWE for 25 years and hasn't figured out how to properly do X by now (where X is some basic, fundamental CS proficiency), it's rather unlikely that person will all of a sudden be better at X in only 6 months or a year.)