The trick is that the Scythians were a nomadic group. Their "homeland" was a wide swath of steppe north of the Black Sea, from the Danube to the Don. They did not settle in one place, but rather roamed around and occasionally raided neighboring territories.

Another problem is that we really don't have much in terms of Scythian texts, so we don't really have their origin story from their point of view, the way we do with other groups. Most of our information comes either from Greek historians such as Herodotus and Hesiod, as well as archaeology. What archaeology tells us is that the Scythians are very similar to (practically indistinguishable from) the Cimmerians, another steppe group that has linguistic ties to the Iranian language tree. Scholars believe the Iranian tie mainly because of the personal and place names of Cimmerian and Scythian people and places that have been recorded in other languages. We can see the names the Assyrians recorded of Cimmerians, and we see that they exhibit typical traits of Iranian languages, just like in modern times we can see the name John MacDonald and know that it's a Scottish name, or David Cohen and know that it's a Jewish name. That's how we figure out the language family of languages we don't have a ton of written evidence for, we compare the personal names we have in other languages and compare them to the language families we do know.

And because we have good evidence from this that the Cimmerians are derived from Iranian-speaking peoples, and good archaeological evidence that the Cimmerians and the Scythians share a very similar cultures, there are a number of very strong ties between the Scythians and the Iranian-speaking peoples.

The last point I want to make is that "Iranian language" does not necessarily mean that they came from the modern geographical region currently occupied by the political entity known as Iran. It's an entire branch of the Indo-European language tree, which includes a lot of languages all over Central and Southern Asia. The specific part of the Indo-Iranian tree that Scythian falls under also includes Bactrian (spoken in ancient Afghanistan, Pakistan, and Tajikistan) and Khwarezmian (spoken in ancient Uzbekistan, Turkmenistan, and Kazakhstan). So it's not actually tied to the geographical area of Iran.

In terms of origins, the exact truth is unclear. There are many opinions because there just isn't enough evidence to be conclusive. There are a lot of Central Asian people groups who spoke Indo-Iranian languages. Where the Scythians were before they became Scythians is unclear. The three origin myths of them that we have are from Herodotus and are contradictory. The two strongest theories are that they either came from the steppes of Inner Asia or came from the area of the Volga-Ural steppes. Either way, I don't think it's accurate to say they came from "Asia" and that term has a lot of connotations with China and Mongolia, and the evidence seems to point to a much more central Asian origin for the Scythians. Same with Germanic and Celtic origins, I don't think the evidence is there. Israelite origins are ridiculous.

If you want to read about it, I recommend The Cambridge History of Early Inner Asia, Volume 1 by Denis Sinor and Empires of the Silk Road: A History of Central Eurasia from the Bronze Age to the Present

Why the origins of the Scythians are so disputed?

They aren't. The web is dark and full of nutters, but the prevailing consensus among academics is that they were Indo-Iranian. Because the Scythians didn't read or write (at least in their own language), we are left with only one source, which is their graves. These graves, in the case of nobility, often bear the names of the entombed in whichever literate language was dominant in the area (the Scythians occupied an enormous part of the steppe, so some graves have names in Old Persian, some in Greek etc). We know they were Indo-Iranian because their names were Indo-Iranian, and they "fit", archeologically speaking, into an area full of other Indo-Iranian groups.

Claiming origin from one nation or the other is a favorite past-time of nationalist nutters, in this particular case it's called Sarmatism. The thing is, the Scythians probably never identified themselves as a single, monolithic ethnic group or nation in the modern sense, but rather they were a tribal confederation composed of many tribes, just like the Huns. The tribal identity was what was important to the everyday man.