We know how far away the Large Magellanic Cloud is because of how bright the stars are as we see them from here. Specifically, stars called Cepheid variables are used, because we can determine their absolute brightness by measuring their period of variation. But in any case, you can tell immediately that the LMC is very far away, simply because of the dimness and angular proximity of its stars. Even young-earth creationists admit that other galaxies are far away; if they were very close by, there'd be no need to postulate that the speed of light has changed.
Enter SN1987a. After it exploded, a shockwave of light and neutrinos went flying away from it. As it propagated outwards, it illuminated interstellar matter, which allows us to measure its progress as it moves perpendicular to our line of sight. IF we postulate that light moved much faster back then, we would see the shockwave propagate outwards very quickly, many light-years per year. Our measurement of the distance to the LMC with Cepheid variables is accurate to maybe 10%, more than good enough to distinguish between a normal speed of light and a vastly greater speed of light. Notice how I haven't said anything about how long ago the explosion occurred: the measurement of the speed of light back then depends only on how far away it was, and how big the shockwave appears to us after 14 years of expansion.
It is an experimentally measured fact that the speed of light in the Large Magellanic Cloud back when SN1897a occurred--whenever it may have occurred--was not significantly different from what it is now. Having conclusively established the stability of the speed of light, we are now at liberty to use it to calculate how long ago it occurred. At a measured one light year per year, that would mean SN1987a occurred 150,000 years ago.