How is rare steak made safe to eat?

First, 145 °F (63 °C) and higher is the temperature for a well done steak. So, with the addition of carry-over cooking, your results don't surprise me. If you are shooting for rare, cook to an internal temperature of 125 °F (52 °C), and let your steak rest 10 minutes before slicing.

While the USDA correctly and necessarily provides temperature guidelines, in fact the reduction of pathogens follows a logarithmic curve and includes the variables of temperature and time. That means, in general, that longer times at lower temperatures will reduce pathogens. This understanding is the basis of sous vide cooking, for example.

Additionally, we generally assume that any potential pathogens are only present on the surface of whole muscle cuts. So, again, in general, achieving the target temperature on the surface eliminates the threat.

Finally, the quality and handling of the raw product is critical. It is important that you have fresh products, kept refrigerated or frozen until use, and handled by people who are practicing safe handling procedures (washed hands or gloves, ...).

One answer I didn't see above: (Edit: Well, it was there. Guess I missed it. Leaving this to cover details that were omitted)

Cooking for safety is a function of temperature and TIME. For example, the USDA recommends 165° for chicken. But, that's an instantaneous temperature read. It's perfectly safe to eat chicken that was cooked to lower temperatures, held for a longer time.

For example, serious eats has the following chart at their food lab guide to sous vide: (Their chart was in-turn derived from a USDA link which is no longer live)

Temperature     Time
136°F (58°C)    68.4 minutes
140°F (60°C)    27.5 minutes
145°F (63°C)    9.2 minutes
150°F (66°C)    2.8 minutes
155°F (68°C)    47.7 seconds
160°F (71°C)    14.8 seconds
165°F (74°C)    Instant

So, if you want medium rare chicken, just bring it up to 140° and hold it at that temperature for 27.5 minutes. Steak is safer for additional reasons also discussed, but it's perfectly possible to have less-than-well done chicken and pork safely.

The USDA recommends cooking many meats

many

That is your problem right there. This USDA guideline is one that, if followed, makes almost all meat safe to eat. Meaning that it caters to the lowest common denominator, the cheapest meat out there.

It's like recommending a complete Hazmat suit with oxygen bottle for anyone working with any chemicals. Yes, everyone will be perfectly fine, but it's an overkill for 99% of activities.

And the same goes for your beef: you're bringing out the hazmat gear for a high schoolers "growing crystals" experiment.

So

So what keeps the raw meat safe enough for us to eat rare?

Generally two things: first of all, its beef as opposed to pork or chicken. The latter two can carry many diseases which spreads to humans. In cows there are very few such diseases and all those are tested for. Secondly: people give a shit. Especially with the meat that decent steakhouses procure, they have their tested and tried supply chains and pay the premium for people to give a shit about hygiene.

In Europe safety regulations means that this care is (supposed to be) taken for all beef but I wouldn't trust it with random stuff from the supermarket. Go to your local butcher and ask them about beef you could eat raw and if they are good they should have some.

Are there things I do or should do to ensure safe raw meat (e.g. sourcing, preservation, preparation, cooking)?

Raw beef from a healthy cow is sterile, so most of the pathogens you could poison yourself with develop on the surface. With that knowledge:

• Smell the meat you're going to eat raw. Spoilage on the surface typically starts with "sour" smell, which gradually turns into "rotten"

• Prefer big chunks of meat over smaller slices

• If in doubt, cut the outer parts of the chunk and cook them conventionally, only consume the inner part raw

• Don't consume mass-produced ground meat raw: it is a mix coming from different animals, so the risk of getting poisoned by the meat from a sick animal is multiplied. Plus, the whole mass is essentially "surface"

Poultry and pork bear much higher risk of inner pathogens and should not be eaten raw.