There isn't a consensus on the origin of saffron, with Greece, Anatolia, and Iran all commonly suggested. What we know is that all three regions were early growers of saffron, but history and archaeology don't tell us which was first.
When and why did saffron production skip over all of Europe from the Middle East and end up in Spain?
Historically, saffron growing spread around the whole Mediterranean, starting from the eastern Mediterranean. Ancient cultivation took place in Anatolia, Greece, Italy, Gaul, Spain, and North Africa, and in most of these regions, there has been a continuous traditional of cultivation from ancient times (saffron cultivation appears to have disappeared from Gaul and been reintroduced into Medieval France, and Albanian cultivation is a modern introduction (or re-introduction?), and also a modern re-introduction in Lebanon). Spain is famous for saffron, having been the main exporter to western Europe for a long time, and Spanish saffron is famed for its quality. Currently, Morocco and Greece outproduce Spain, but their saffron doesn't enjoy the reputation of that of Spain. Italy and Egypt are also significant producers, but in most of the other Mediterranean countries, production is relatively small. Spanish saffron is also famed for its quality. Saffron has also been grown more widely in Europe in the past (e.g., England was a significant producer), and is still grown commercially in Austria and, on a small scale, Switzerland. (And, of course, more widely to the east as well, with India being a major producer (currently 2nd after Iran), and China also significant, but this eastern production isn't relevant to the Mediterranean question.)
The fame of Spanish saffron is such that it overshadows other Mediterranean production, to the point of other Mediterranean countries not appearing on many abbreviated lists of producers. The question to be asked isn't why saffron production skipped the rest of the Mediterranean - because it didn't - but why Spain became such a dominant and famous exporter to western Europe and the rest of the West, while other regions produced largely for the domestic market and/or failed to achieve Spain's reputation for quality.
The main point here is that saffron didn't really skip anywhere. It is unknown exactly where saffron originated from, though as u/wotan_weevil notes, it was most likely Iran, Greece or Asia Minor- and by the time of the Roman empire it was being traded throughout and produced in several parts of the Mediterranean basin. Pliny the Elder in his Natural History mused on the best saffrons: "The most esteemed saffron, however, is that of Cilicia (southeast Asia Minor), and there of Mount Corycus in particular; next comes the saffron of Mount Olympus, in Lycia, and then of Centuripa, in Sicily." Pliny goes on to explain how saffron is so often adulterated, and how to recognize the best quality (because some things never change).
Apicius, who wrote the only extant Roman cookery book, includes saffron in a recipe for sauce for wild boar: "Crush pepper, cumin, celery seed, mint, thyme, satury, saffron, toasted nuts, or toasted almonds, honey, wine, broth, vinegar and a little oil."
Saffron was already valued in antiquity for its potential as a food, medicine, dye, and as an aromatic for perfume- and its low weight and high cost made it a perfect trade good. What is, admittedly, absent, is any direct archeological or textual evidence that proves saffron cultivation in Spain- though as an important part of Phoenician, Greek, and later Roman trade routes, it is reasonable to believe that Spain would have had access to saffron and saffron products, two thousand years ago. But with the fall of the Roman empire in the west, trade across the Mediterranean declined. And then very little is known about the diet of the Visigoths in Spain- the best primary source is Isidore of Seville (hopeful patron saint of the internet), who in the seventh century described the local diet as rather simple, based in cereals, milk and cheese (from sheep), and wine. If there was saffron production in Spain, he doesn't mention it either.
Finally, we get to the traditional answer to this question within culinary historiography- saffron, as well as a variety of other foods including rice, oranges/lemons, rose water, and sugar was brought to Spain as a result of the Moorish/Arab conquest, which started in 711 AD. They established the kingdom of Al-Andalus in Spain, expanded irrigation and cultivation (especially in southern and eastern Spain) and made saffron a permanent part of the Spanish diet- if it wasn't already (which is possible but impossible to prove). The same thing happened to Sicily when it was invaded in 827, although we know that Sicily already had a history of saffron cultivation. The spread of Islam linked Spain to Persia and beyond throughout much of the Middle Ages, and made it a part of trade networks that spread all the way to East Asia and the spice islands.