The way the FBI "determined" that Russia hacked the DNC's server was basically that the DNC said so. It was a little more complicated than that: the DNC produced a report from their contractor Crowdstrike, large portions of which were redacted. This imagined court room exchange serves to describe how that might play out when exposed to legal scrutiny:

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DEFENSE ATTORNEY: Agent Smith, you testified that the Russians hacked the DNC computers, is that correct?

FBI AGENT JOHN SMITH: That is correct.

DEF ATT: Upon what information did you base your testimony?

AGENT: Information found in reports analyzing the breach of the computers.

DEF ATT: So, the FBI prepared these reports?

AGENT: (cough).... (shift in seat) No, a cyber security contractor with the FBI.

DEF ATT: Pardon me, why would a contractor be preparing these reports?

Do these contractors run the FBI laboratories where the server was examined?

AGENT: No.

DEF ATT: No? No what? These contractors don’t run the FBI Laboratories?

AGENT: No. The laboratories are staffed by FBI personnel.

DEF ATT: Well I don’t understand.Why would contractors be writing reports about computers that are forensically examined in FBI laboratories?

AGENT: Well, the servers were not examined in the FBI laboratory.

(silence)

DEF ATT: Oh, so the FBI examined the servers on site to determine who had hacked them and what was taken?

AGENT: Uh..... no.

DEF ATT: They didn’t examine them on site?

AGENT: No.

DEF ATT: Well, where did they examine them?

AGENT: Well, uh..... the FBI did not examine them.

DEF ATT: What?

AGENT: The FBI did not directly examine the servers.

DEF ATT: Agent Smith, the FBI has presented to the Grand Jury and to this court and SWORN AS FACT that the Russians hacked the DNC computers. You are basing your SWORN testimony on a report given to you by a contractor, while the FBI has NEVER actually examined the computer hardware?

AGENT: That is correct.

DEF ATT: Agent Smith, who prepared the analysis reports that the FBI relied on to give this sworn testimony?

AGENT: Crowdstrike, Inc.

DEF ATT: So, which Crowdstrike employee gave you the report?

AGENT: We didn’t receive the report directly from Crowdstrike.

DEF ATT: What?

AGENT: We did not receive the report directly from Crowdstrike.

DEF ATT: Well, where did you find this report?

AGENT: It was given to us by the people who hired Crowdstrike to examine and secure their computer network and hardware.

DEF ATT: Oh, so the report was given to you by the technical employees for the company that hired Crowdstrike to examine their servers?

AGENT: No.

DEF ATT: Well, who gave you the report?

AGENT: Legal counsel for the company that hired Crowdstrike.

DEF ATT: Why would legal counsel be the ones giving you the report?

AGENT: I don’t know.

DEF ATT: Well, what company hired Crowdstrike?

AGENT: The Democratic National Committee.

DEF ATT: Wait a minute. Let me get this straight. You are giving SWORN testimony to this court that Russia hacked the servers of the Democratic National Committee. And you are basing that testimony on a report given to you by the LAWYERS for the Democratic National Committee and you, the FBI, never actually saw or examined the computer servers?

AGENT: That is correct.

DEF ATT: Well, can you provide a copy of the technical report produced by Crowdstrike for the Democratic National Committee?

AGENT: No, I cannot.

DEF ATT: Well, can you go back to your office and get a copy of the report?

AGENT: No.

DEF ATT: Why? Are you locked out of your office?

AGENT: No.

DEF ATT: I don’t understand. Why can you not provide a copy of this report?

AGENT: Because I do not have a copy of the report.

DEF ATT: Did you lose it?

AGENT: No.

DEF ATT: Why do you not have a copy of the report?

AGENT: Because we were never given a final copy of the report.

DEF ATT: Agent Smith, if you didn’t get a copy of the report, upon what information are you basing your testimony?

AGENT: On a draft copy of the report.

DEF ATT: A draft copy?

AGENT: Yes.

DEF ATT: Was a final report ever delivered to the FBI?

AGENT: No.

DEF ATT: Agent Smith, did you get to read the entire report?

AGENT: No.

DEF ATT: Why not?

AGENT: Because large portions were redacted.

DEF ATT: Agent Smith, let me get this straight. The FBI is claiming that the Russians hacked the DNC servers. But the FBI never actually saw the computer hardware, nor examined it? Is that correct?

AGENT: That is correct.

DEF ATT: And the FBI never actually examined the log files or computer email or any aspect of the data from the servers? Is that correct?

AGENT: That is correct.

DEF ATT: And you are basing your testimony on the word of Counsel for the Democratic National Committee, the people who provided you with a REDACTED copy of a DRAFT report, not on the actual technical personnel who supposedly examined the servers?

AGENT: That is correct.

DEF ATT: Your honor, I have a few motions I would like to make at this time.

PRESIDING JUDGE: I’m sure you do, Counselor. (as he turns toward the prosecutors) And I feel like I am in a mood to grant them.
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The above was written some time ago by an anon on 8chan. The original post is now lost, but a screen grab of it was posted today on 8kun:
https://8kun.top/qresearch/res/8066064.html#8066472

I found a text version which had been reposted on Twitter, which was my source for the above:
https://threader.app/thread/1177176058958770177