Oh-my-god particle: How can it get through Milky way in 10 seconds?

Key point in your quote is: "from protons reference frame". In the reference frame, travelling at a relativistic speed, length contraction is experienced. All the lengths in the direction of travel of the particle are contracted by Lorentz factor: $$ l'=\frac{l}{\gamma}$$ $$ \gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$$ So $ \gamma = \frac{1}{\sqrt{1-(0.9999999999999999999999951})^2}=3.19*10^{11}. $ In the reference frame of the particle, Milkyway is contracted by this factor. So the proton sees it only $2.96 * 10^9 m$ long. Now you can do the usual calculation to find time using the new contracted length and see that it would take only $2.96 * 10^9/ 3*10^8 = 9$ seconds to cross the Milkyway.

Length contraction is kind of a consequence of 4D space-time we live in. If you look at time dilation (which also can be used to derive this result but is less intuitive in my opinion), the length contraction naturally arises from it. If you want to know more about length contraction you can easily more information on it. It is a topic which is usually well explained in any special relativity book,and I bet there are a lot of question on the topic on this website, search tags length-contraction and special-relativity.

Relativity makes time relative (what a surprise! :)). It makes a difference, whether we look at the particle from "outside", or if we travel along with the same speed.

So viewed from outside the particle is a normal fast particle, and need some hundreds of thousands of years for the milky way. The "oh-my-god"-ity does nothing here, it makes it only a few seconds faster that plenty of others.

Only the particle itself doesn't experience it like this. You can either say, that time flows much slower in the frame of the particle, or that the milky way is much shorter (again, as seen by the particle only!).

These are two equivalent formulations, and you will find them explained in more breadth on the first pages of every relativity book.

There is a common misunderstanding about special relativity related to this. What we in physics call a velocity is distance/time where distance and time are measured by the same observer. This one is always smaller than the speed of light. So... if we measure the galaxy and we measure the time, we see the particle needing quite a lot of time to get from there to here.

What the particle itself sees is a VERY relativistically contracted space. The galaxy for the particle looks like a bright (doppler-shifted) and extremely thin pancake. So naturally, it takes it almost no time to travel across. That's the 10 seconds the article is talking about. That is, if the particle itself had a watch on its wrist, it would measure that time. Again, dividing these two quantities gives you the hairline below speed of light.

But the speed "limit" doesn't mean you can't get very far in your own time. It actually makes travel easier. If you say things naively: measure the distance in earth coordinates, then accelerate and measure your own time while traveling, and divide the two, you can get any number you want, this isn't really a physical quantity at all, the quantities are measured from different reference frames and what you get isn't really a velocity. You can get anywhere you want in almost no time (sure, people on earth will be long dead when you get there, but for you it'll be a short journey).