How do you explain the principal of "you cannot prove a negative" to someone that is very weak at abstractions like this?

Depends somewhat on the particular negative. We can claim with certainty that dinosaurs are not alive today, even though there are theoretically places that are not sufficiently explored where, in principle, a dinosaur might live. We are well aware that the fossil record is globally consistent and demonstrates the timing of their extinction, as well as the lack of any fossils of dinosaurs since then.
Similarly, I can claim with confidence that a force does not exist in the universe that would allow me to teleport from place to place instantaneously, because such a force would violate many of the known laws of physics.
There are various types of god-beings that cannot exist because of a logical contradiction. An omnipresent entity, for instance, would be always detectable if it had any measurable effect on the world, but no such measurable effect can be observed. If it is not capable of a measurable effect, then it is indistinguishable from not existing.
Other types of negatives are not easily proven. That is why scientists will say things like, “Based on 3478 measurements, there is currently no evidence to support the existence of higher values than those reported here.”
- Maybe we don’t detect the omnipresent being because we ALWAYS detect it, but as a constant?

Prove to me you never raped your niece
Be careful with this one though.

By simple analogy. You can prove that there are white crows by finding a single white crow, but to prove that there are no white crows you must conduct an exhaustive search of every corner of the earth and never find a single one and somehow be absolutely certain that you didn't miss one somewhere.
The only way to be absolutely certain that you didn't miss something is to be able to look everywhere all at once, otherwise a white crow might evade your notice, and that's impossible.
As such all you can say is there probably aren't any white crows because we have lots of experience seeing crows and there has been no evidence of one yet.
- This is a very good analogy!
  Although to be pedantic, there are white crows; there was one living in my neighborhood some years back.
  
  It's just an example. Mentally edit it to polka-dotted hippopotamuses if it makes ya happy. ;)
- Similar to the argument Intelligent Design (anti evolutionists) use. They always point to fossil gaps and say "Well, what happened there?" You have no control over what gets fossilized and where you can find them and there will always be a gap somewhere. "Well there is a gap on Tuesday, 67,000,000 BC!" There is no possible way to fill in every imaginable gap with physical evidence vs seeing patterns and development across those gaps.
- As such all you can say is there probably aren’t any white crows because we have lots of experience seeing crows and there has been no evidence of one yet.
  
  It's just an example man.

The invisible pink unicorn is the traditional atheist approach.
- tl;dr for others
  A: "I have an Pink Unicorn inside the trunk of my car. It vanishes the moment you try to open the trunk or look at it."
  B: "What? That's absurd"
  A: "I know it exists. It's up to you to disprove it"
  B: "But there is no way one can capture/observe/understand it with any sort of scientific instrument"
  A: "Don't care. Skill issue"
  
  Jokes on you. I drive a wrangler. No trunk.

Stop talking shit about me.
I'm not.
Oh yeah? prove it.

If you subscribe to classical logic (i.e., propositonal or first order logic) this is not true. Proof by contradiction is one of the more common classical logic inference rules that lets you prove negated statements and more specifically can be used to prove nonexistence statements in the first order case. People go so far as to call the proof by contradiction rule "not-introduction" because it allows you to prove negated things.
Here's a wiki page that also disagrees and talks more specifically about this "principle": source (note the seven separate sources on various logicians/philosophers rejecting this "principle" as well).
If you're talking about some other system of logic or some particular existential claim (e.g. existence of god or something else), then I've got not clue. But this is definitely not a rule of classical logic.
- When people colloquially say "you cannot prove a negative" they are usually referring to the fact that absence of evidence can not be used to deduce non-existence of some phenomena ("a negative"), whereas the factual discovery of a phenomena can be used to deduce that the phenomena exists ("a positive").
  They are therefore not referring to formal negation but rather making a point about deductive vs. inductive reasoning and the asymmetry of these two related questions (existence vs. nonexistence).
  There is a bit of nuance to add here in that practically speaking you can't really "discover a fact" by direct observation. But again this is a colloquialism as most laypeople will accept what is directly observable under their noses as factual rather than a noisy data point of one.
  
  I think you are assuming a level of competence from people that I don't have faith people actually have. People absolutely can and do take "you cannot prove a negative" as a real logical rule in the literal negation sense. This isn't colloquialism. This is people misunderstanding what the phrase means.
  I have definitely had conversations with idiots that have taken this phrase to mean that you just literally cannot logically prove negated statements. Whether folks like you get that that is not what the phrase refers to is irrelevant to why I'm pointing out the distinction.

Are you talking to a conspiracy nut? If so, you can forget about reasoning with them. They don’t play this game by the same rules as you do.
No amount of logic, facts or evidence will ever help. They have emotional issues, so you need to use an emotional solution.
- Convincing someone to abandon their tribe won’t be easy. No matter how logical your explanation is, they’re going to stick to their tribe regardless.

Russels teapod is a good example imho
EDIT: https://en.m.wikipedia.org/wiki/Russell%27s_teapot
- He wrote that if he were to assert, without offering proof, that a teapot, too small to be seen by telescopes, orbits the Sun somewhere in space between the Earth and Mars, he could not expect anyone to believe him solely because his assertion could not be proven wrong.
  I love that one of the arguments against this analogy, shown in the Wikipedia, article is the following:
  the only way a teapot could have gotten into orbit around the sun would be if some country with sufficiently developed space-shot capabilities had shot this pot into orbit. No country with such capabilities is sufficiently frivolous to waste its resources by trying to send a teapot into orbit.
  ... what a time to be alive!
  
  Dude...imagine if we could convince Trump/Musk and Space Force/Space X to do this. It's like philosophy's version of the Torment Nexus!
  
  A teapot, no. A racecar, sure, did it a while back and then basically stopped talking about it.